Limits built on sin(u)/u → 1 and (1 − cos u)/u → 0. 69 problems with worked solutions; in 67 of them every
equation is proved by a computer algebra system.
20 worked problems
Try each one before opening the solution. Every line is labeled with what checked it; 20 of these also have a second solution, written separately and checked the same way. How we verify
limit-lawApply the product rule for limits.✓ Proved
\[ = 1 \]
limitsimplifysimplifyEvaluate the first limit component. Evaluate the second limit component. Final result.✓ Proved
Answer \( 1 \)
The graph of f(x) = sin(2x + 1)²/(2x + 1)². Two points slide along the curve toward x = -1/2, one from each side; their heights close in on 1.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\sin^{2}{\left(2 x + 1 \right)}}{\left(2 x + 1\right)^{2}}\right) \]
limitrewriteRewrite the fraction as a square of sin over its argument.✓ Proved
\[ = \left(\lim_{x \to - \frac{1}{2}^+}\left(\frac{\sin{\left(2 x + 1 \right)}}{2 x + 1}\right)\right)^{2} \]
limit-lawApply the limit law to the inner fraction.✓ Proved
\[ = 1 \]
limitsimplifySince sin(u)/u -> 1 as u -> 0. The limit evaluates to 1.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
limitlimitvalue of inner limit is 1 final value✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
lhopitalsimplifyCompute the derivatives of the numerator and denominator. Simplify the constant fraction.✓ Proved
\[ = \frac{4}{3} \]
limitsimplifysimplifysimplifyEvaluate the limit by substituting x = -1/2. Simplify the argument of the cosine function. Evaluate the cosine of zero. Final value.✓ Proved
Answer \( \frac{4}{3} \)
The graph of f(x) = sin(8x + 4)/(6x + 3). Two points slide along the curve toward x = -1/2, one from each side; their heights close in on 4/3.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\sin{\left(8 x + 4 \right)}}{6 x + 3}\right) \]
limit✓ Proved
\[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \sin{\left(8 x + 4 \right)}}{\frac{d}{d x} \left(6 x + 3\right)}\right) \]
simplifysimplifyCompute the derivatives Reduce the fraction✓ Proved
\[ = \frac{4}{3} \]
limitEvaluate the limit✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
algebraFactor out the constant 3 from the denominator.✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{\sin^{2}{\left(3 u \right)}}{9 u^{2}}\right) \]
substitutionsimplifyLet u = x - 1/3, so u approaches 0. Simplify the expression inside the limit.✓ Proved
\[ = \lim_{u \to 0^+} 1 \]
trig-limitsimplifyUse the fundamental trigonometric limit sin(u)/u -> 1. Evaluate the constant expression.✓ Proved
\[ = 1 \]
limitThe limit evaluates to 1.✓ Proved
Answer \( 1 \)
The graph of f(x) = sin(3x - 1)²/(3x - 1)². Two points slide along the curve toward x = 1/3, one from each side; their heights close in on 1.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\sin^{2}{\left(3 x - 1 \right)}}{\left(3 x - 1\right)^{2}}\right) \]
limitfactorRewrite the fraction as a square of sin(u)/u.✓ Proved
\[ = \left(\lim_{x \to \frac{1}{3}^+}\left(\frac{\sin{\left(3 x - 1 \right)}}{3 x - 1}\right)\right)^{2} \]
limit-lawApply the limit law to separate the square.✓ Proved
\[ = 1 \]
limitsimplifySince sin(u)/u → 1 as u → 0, and 3*x-1 → 0 at x=1/3.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{\sin{\left(15 x - 5 \right)}}{\sin{\left(6 x - 2 \right)}} \).
\[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\sin{\left(15 x - 5 \right)}}{\sin{\left(6 x - 2 \right)}}\right) \]
limitIdentify the limit and the indeterminate form.✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \sin{\left(15 x - 5 \right)}}{\frac{d}{d x} \sin{\left(6 x - 2 \right)}}\right) \]
lhopitalApply L'Hopital's Rule because the limit is of the form 0/0.✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{5 \cos{\left(15 x - 5 \right)}}{2 \cos{\left(6 x - 2 \right)}}\right) \]
simplifyCompute the derivatives.✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+} \frac{5}{2} \]
limitEvaluate the limit by substituting x = 1/3.✓ Proved
\[ = \frac{5}{2} \]
algebrasimplifysimplifysimplifysimplifySimplify the arguments of the trigonometric functions. Evaluate the arithmetic inside the cosine functions. Substitute the value of cos(0). Simplify the fraction. Final simplified value.✓ Proved
Answer \( \frac{5}{2} \)
The graph of f(x) = sin(15x - 5)/sin(6x - 2). Two points slide along the curve toward x = 1/3, one from each side; their heights close in on 5/2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\sin{\left(15 x - 5 \right)}}{\sin{\left(6 x - 2 \right)}}\right) \]
limitOriginal limit✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \sin{\left(15 x - 5 \right)}}{\frac{d}{d x} \sin{\left(6 x - 2 \right)}}\right) \]
lhopitalApply l'Hôpital's rule✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{5 \cos{\left(15 x - 5 \right)}}{2 \cos{\left(6 x - 2 \right)}}\right) \]
simplifyCompute derivatives✓ Proved
\[ = \frac{5}{2} \]
limit-lawlimitEvaluate at x = 1/3 Final value✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
lhopitalApply L'Hôpital's rule because the limit is an indeterminate form 0/0.✓ Proved
\[ = \lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{6 x}\right) \]
simplifyalgebrarewriteDifferentiate the numerator and denominator. Simplify the fraction by dividing the constants. Rewrite the expression to use the standard trigonometric limit.✓ Proved
\[ = \frac{\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{3 x}\right)}{2} \]
limit-lawApply the limit-law for the product of two functions.✓ Proved
\[ = \frac{1}{2} \]
trig-limitsimplifyUse the standard limit result sin(u)/u -> 1 as u -> 0. Final evaluation.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (1 - cos(3x))/(9x²). Two points slide along the curve toward x = 0, one from each side; their heights close in on 1/2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to 0^+}\left(\frac{1 - \cos{\left(3 x \right)}}{9 x^{2}}\right) \]
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
limit-lawrewritePull out the constant factor 3. Rewrite the sine argument as 3*(x+1/3).✓ Proved
\[ = 3 \lim_{t \to 0^+}\left(\frac{t}{\sin{\left(3 t \right)}}\right) \]
substitutionLet t = x + 1/3, so t -> 0.✓ Proved
\[ = 3 \lim_{t \to 0^+} \frac{1}{3} \]
limit-lawUse sin(3*t) ~ 3*t near 0.✓ Proved
\[ = 1 \]
limitlimitEvaluate the remaining constant limit. Final value of the limit.✓ Proved
✓ Nihil obstat Lines: 7 proved, 2 checked by independent solution. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
lhopitalApply L'Hopital's rule to the inner limit.✓ Proved
\[ = \left(\lim_{x \to 0^+} \cos{\left(3 x \right)}\right)^{2} \]
simplifysimplifyCompute the derivatives. Simplify the fraction inside the limit.✓ Proved
\[ = 1 \]
limitsimplifyEvaluate the limit of the cosine function. Final result.✓ Proved
Answer \( 1 \)
The graph of f(x) = sin(3x)²/(9x²). Two points slide along the curve toward x = 0, one from each side; their heights close in on 1.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(3 x \right)}}{9 x^{2}}\right) \]
limitrewriterewritefactor out 9*x**2 expand the square✓ Proved
\[ = \left(\lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{3 x}\right)\right)^{2} \]
limit-lawproduct of limits✓ Proved
\[ = \lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{3 x}\right) \]
limitsimplify one factor✓ Proved
\[ = 1 \]
limitalgebrasimplifyknown limit sin(u)/u → 1 multiply the two limits evaluate the product✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Evaluate \( \displaystyle \lim_{x \to -1} \frac{\sin{\left(5 x + 5 \right)}}{\sin{\left(2 x + 2 \right)}} \).
\[ \lim_{x \to -1^+}\left(\frac{\sin{\left(5 x + 5 \right)}}{\sin{\left(2 x + 2 \right)}}\right) \]
limitfactorEvaluate the limit of the given function. Factor the arguments of the sine functions.✓ Proved
\[ = \lim_{x \to -1^+}\left(\frac{\frac{d}{d x} \sin{\left(5 x + 5 \right)}}{\frac{d}{d x} \sin{\left(2 x + 2 \right)}}\right) \]
lhopitalThe limit is an indeterminate form 0/0, so apply L'Hopital's rule.✓ Proved
\[ = \lim_{x \to -1^+}\left(\frac{5 \cos{\left(5 x + 5 \right)}}{2 \cos{\left(2 x + 2 \right)}}\right) \]
simplifyCompute the derivatives of the numerator and denominator.✓ Proved
\[ = \lim_{x \to -1^+} \frac{5}{2} \]
substitutionsimplifysimplifySubstitute the value x = -1 into the expression. Evaluate the cosine terms. Simplify the numerical fraction.✓ Proved
\[ = \frac{5}{2} \]
limitThe limit is evaluated.✓ Proved
Answer \( \frac{5}{2} \)
The graph of f(x) = sin(5x + 5)/sin(2x + 2). Two points slide along the curve toward x = -1, one from each side; their heights close in on 5/2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to -1^+}\left(\frac{\sin{\left(5 x + 5 \right)}}{\sin{\left(2 x + 2 \right)}}\right) \]
limitrewriteFactor out (x+1) from the arguments of the sines.✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(5 u \right)}}{\sin{\left(2 u \right)}}\right) \]
substitutionLet u = x + 1, so u -> 0 as x -> -1.✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(5 u \right)}}{2 \sin{\left(u \right)}}\right) \]
limit-lawRewrite sin(2*u) as 2*sin(u).✓ Proved
\[ = \frac{\lim_{u \to 0^+}\left(\frac{\sin{\left(5 u \right)}}{\sin{\left(u \right)}}\right)}{2} \]
limit-lawSeparate the constant factor 1/2.✓ Proved
\[ = \frac{5}{2} \]
limitAs u -> 0, sin(5*u)/sin(u) -> 5.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
simplifysimplifyDifferentiate the numerator and denominator. Simplify the fraction.✓ Proved
\[ = 0 \]
substitutionalgebrasimplifylimitSubstitute the value x = 1/3 into the expression. Simplify the expression inside the sine function. Evaluate the expression inside the sine. The final value of the limit is 0.✓ Proved
Answer \( 0 \)
The graph of f(x) = (1 - cos(3x - 1))/(3x - 1). Two points slide along the curve toward x = 1/3, one from each side; their heights close in on 0.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to \frac{1}{3}^+}\left(\frac{1 - \cos{\left(3 x - 1 \right)}}{3 x - 1}\right) \]
limitlimitEvaluate each factor Final value✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{3}} \frac{\tan{\left(3 x + 1 \right)}}{3 x + 1} \).
\[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\tan{\left(3 x + 1 \right)}}{3 x + 1}\right) \]
limitalgebrasimplifySet up the limit. Rewrite the denominator to isolate the term approaching zero. The expression is already in a form suitable for substitution.✓ Proved
substitutionLet u = 3*x + 1; then x -> -1/3 gives u -> 0.✓ Proved
\[ = 1 \]
limit-lawThe standard limit lim_{u->0} tan(u)/u = 1.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
simplifyCompute the derivatives of the numerator and denominator.✓ Proved
\[ = \frac{4}{3} \]
limitsimplifyEvaluate the limit by substituting x = 1. Simplify the final numerical value.✓ Proved
Answer \( \frac{4}{3} \)
The graph of f(x) = sin(4x - 4)/(3x - 3). Two points slide along the curve toward x = 1, one from each side; their heights close in on 4/3.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to 1^+}\left(\frac{\sin{\left(4 x - 4 \right)}}{3 x - 3}\right) \]
limitrewriteRewrite 4*x-4 as 4*(x-1).✓ Proved
\[ = \lim_{x \to 1^+}\left(\frac{4 \sin{\left(4 x - 4 \right)}}{3 \left(4 x - 4\right)}\right) \]
factorFactor out 4 from the denominator.✓ Proved
\[ = \frac{4 \lim_{x \to 1^+}\left(\frac{\sin{\left(4 x - 4 \right)}}{4 x - 4}\right)}{3} \]
limit-lawSeparate constant factor.✓ Proved
\[ = \frac{4}{3} \]
limitsimplifyUse lim_{u->0} sin(u)/u = 1 with u = 4*(x-1). Final value of the limit.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{2}} \frac{\sin{\left(10 x + 5 \right)}}{\sin{\left(4 x + 2 \right)}} \).
\[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\sin{\left(10 x + 5 \right)}}{\sin{\left(4 x + 2 \right)}}\right) \]
limitfactorSet up the limit. Factor the arguments of the sine functions.✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(5 u \right)}}{\sin{\left(2 u \right)}}\right) \]
substitutionLet u = 2*x + 1, so u approaches 0 as x approaches -1/2.✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} \sin{\left(5 u \right)}}{\frac{d}{d u} \sin{\left(2 u \right)}}\right) \]
lhopitalApply L'Hôpital's rule since the limit is an indeterminate form 0/0.✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{5 \cos{\left(5 u \right)}}{2 \cos{\left(2 u \right)}}\right) \]
simplifyCompute the derivatives.✓ Proved
\[ = \frac{5}{2} \]
limitsimplifyEvaluate the limit by substituting u = 0. Simplify the final value.✓ Proved
Answer \( \frac{5}{2} \)
The graph of f(x) = sin(10x + 5)/sin(4x + 2). Two points slide along the curve toward x = -1/2, one from each side; their heights close in on 5/2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\sin{\left(10 x + 5 \right)}}{\sin{\left(4 x + 2 \right)}}\right) \]
limit✓ Proved
\[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \sin{\left(10 x + 5 \right)}}{\frac{d}{d x} \sin{\left(4 x + 2 \right)}}\right) \]
lhopital✓ Proved
\[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{5 \cos{\left(10 x + 5 \right)}}{2 \cos{\left(4 x + 2 \right)}}\right) \]
limit✓ Proved
\[ = \frac{5}{2} \]
simplifylimit✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
substitutionalgebraalgebraLet u = x + 1 Factor u/2 out of the denominator Simplify the constant factor✓ Proved
\[ = \lim_{u \to 0^+} \frac{1}{2} \]
limitUse lim_{u->0} sin(u/2)/(u/2) = 1✓ Proved
\[ = \frac{1}{2} \]
limit✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
simplifyCompute the derivatives of the numerator and denominator.✓ Proved
\[ = \frac{4}{3} \]
limitsimplifyEvaluate the limit by substituting x = 1/2. Simplify the final numerical expression.✓ Proved
Answer \( \frac{4}{3} \)
The graph of f(x) = sin(8x - 4)/(6x - 3). Two points slide along the curve toward x = 1/2, one from each side; their heights close in on 4/3.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to \frac{1}{2}^+}\left(\frac{\sin{\left(8 x - 4 \right)}}{6 x - 3}\right) \]
limitrewritefactorRewrite the sine argument. Factor 3*(2*x-1) as 6*(x-1/2).✓ Proved
\[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{4 \sin{\left(8 x - 4 \right)}}{3 \left(8 x - 4\right)}\right) \]
rewriteIntroduce 8*(x-1/2) to use the sin(u)/u limit.✓ Proved
\[ = \frac{4 \lim_{x \to \frac{1}{2}^+}\left(\frac{\sin{\left(8 x - 4 \right)}}{8 x - 4}\right)}{3} \]
limit-lawSeparate constant factor.✓ Proved
\[ = \frac{4}{3} \]
limitsimplifyLimit of sin(u)/u as u→0 is 1. Simplify the constant.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
substitutionLet u = 3*x - 1, so u -> 0 as x -> 1/3.✓ Proved
\[ = 1 \]
limit-lawThe standard limit lim_{u->0} tan(u)/u = 1.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
simplifyCompute the derivatives of the numerator and denominator.✓ Proved
\[ = \frac{4}{3} \]
limitsimplifyEvaluate the limit by substituting x = -1. Simplify the final numerical value.✓ Proved
Answer \( \frac{4}{3} \)
The graph of f(x) = sin(4x + 4)/(3x + 3). Two points slide along the curve toward x = -1, one from each side; their heights close in on 4/3.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to -1^+}\left(\frac{\sin{\left(4 x + 4 \right)}}{3 x + 3}\right) \]
limitrewriteInitial limit Rewrite the argument of sin✓ Proved
\[ = \lim_{x \to -1^+}\left(\frac{4 \sin{\left(4 x + 4 \right)}}{3 \left(4 x + 4\right)}\right) \]
limit-lawFactor out 4/3 using limit law✓ Proved
\[ = \frac{4 \lim_{x \to -1^+}\left(\frac{\sin{\left(4 x + 4 \right)}}{4 x + 4}\right)}{3} \]
limitSeparate the constant 4/3✓ Proved
\[ = \frac{4}{3} \]
limitlimitLimit of sin(u)/u as u→0 is 1 Final value of the limit✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify