Indeterminate forms 0/0 and ∞/∞, settled by differentiating top and bottom. 76 problems with worked solutions; in 72 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
substitutionsimplifySubstitute u = x - 1/3, which implies u approaches 0 as x approaches 1/3. Simplify the denominator (3 * (3*u)**2 = 27*u**2).✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(3 u \right)}}{18 u}\right) \]
simplifyalgebraCompute the derivatives. Simplify the coefficients.✓ Proved
\[ = \lim_{u \to 0^+} \frac{1}{6} \]
trig-limitUse the standard limit sin(kz)/z = k as z approaches 0.✓ Proved
\[ = \frac{1}{6} \]
limitFinal evaluation of the limit.✓ Proved
Answer \( \frac{1}{6} \)
The graph of f(x) = (3x - sin(3x - 1) - 1)/(3x - 1)³. Two points slide along the curve toward x = 1/3, one from each side; their heights close in on 1/6.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to \frac{1}{3}^+}\left(\frac{3 x - \sin{\left(3 x - 1 \right)} - 1}{\left(3 x - 1\right)^{3}}\right) \]
limit✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+} \frac{1}{6} \]
rewritealgebrasimplifysimplifyUse sin(y) ≈ y - y³/6 near y=0 Distribute the minus sign Combine like terms in the numerator Cancel (3*x-1)**3✓ Proved
\[ = \frac{1}{6} \]
limitThe limit equals the constant 1/6✓ 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
trig-limitsimplifysimplifyUse the small angle approximation sin(theta) ~ theta for the limit. Expand the squared term. Cancel the u**2 terms and simplify the constant.✓ Proved
\[ = \frac{1}{6} \]
limitThe limit evaluates to the constant value.✓ Proved
Answer \( \frac{1}{6} \)
The graph of f(x) = (3x - sin(3x + 1) + 1)/(3x + 1)³. Two points slide along the curve toward x = -1/3, one from each side; their heights close in on 1/6.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{3 x - \sin{\left(3 x + 1 \right)} + 1}{\left(3 x + 1\right)^{3}}\right) \]
limitalgebra✓ Proved
\[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(3 x - \sin{\left(3 x + 1 \right)} + 1\right)}{\frac{d}{d x} \left(3 x + 1\right)^{3}}\right) \]
lhopital✓ Proved
\[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{3 - 3 \cos{\left(3 x + 1 \right)}}{9 \left(3 x + 1\right)^{2}}\right) \]
✓ 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
simplifysimplifyCalculate the derivatives. Simplify the resulting fraction.✓ Proved
\[ = \frac{1}{2} \]
limitEvaluate the limit as u approaches 0.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (-2x + e^(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 1/2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{- 2 x + e^{2 x + 1} - 2}{\left(2 x + 1\right)^{2}}\right) \]
limit✓ Proved
\[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(- 2 x + e^{2 x + 1} - 2\right)}{\frac{d}{d x} \left(2 x + 1\right)^{2}}\right) \]
lhopital✓ Proved
\[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{2 e^{2 x + 1} - 2}{8 x + 4}\right) \]
simplify✓ Proved
\[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(2 e^{2 x + 1} - 2\right)}{\frac{d}{d x} \left(8 x + 4\right)}\right) \]
lhopital✓ Proved
\[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{e^{2 x + 1}}{2}\right) \]
simplifysimplify✓ 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
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{- 3 x + e^{3 x - 1}}{\left(3 x - 1\right)^{2}} \).
\[ \lim_{x \to \frac{1}{3}^+}\left(\frac{- 3 x + e^{3 x - 1}}{\left(3 x - 1\right)^{2}}\right) \]
limitalgebraDefine the limit from the right. Rearrange the numerator.✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(- 3 x + e^{3 x - 1}\right)}{\frac{d}{d x} \left(3 x - 1\right)^{2}}\right) \]
lhopitalApply L'Hopital's rule since the limit is 0/0.✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{3 e^{3 x - 1} - 3}{18 x - 6}\right) \]
simplifyalgebrafactorCompute the derivatives. Simplify the denominator. Factor out 3 from the numerator.✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{e^{3 x - 1} - 1}{6 x - 2}\right) \]
algebraSimplify the fraction.✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(e^{3 x - 1} - 1\right)}{\frac{d}{d x} \left(6 x - 2\right)}\right) \]
lhopitalApply L'Hopital's rule again.✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{e^{3 x - 1}}{2}\right) \]
simplifyalgebraalgebraCompute the derivatives. Simplify the fraction. Reduce the fraction.✓ Proved
\[ = \frac{1}{2} \]
limitalgebrasimplifyEvaluate the limit by substitution. Simplify the exponent. Final evaluation.✓ Proved
Answer \( \frac{1}{2} \)
The graph of f(x) = (-3x + e^(3x - 1))/(3x - 1)². Two points slide along the curve toward x = 1/3, 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.
\[ \lim_{x \to \frac{1}{3}^+}\left(\frac{- 3 x + e^{3 x - 1}}{\left(3 x - 1\right)^{2}}\right) \]
limit✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{3 e^{3 x - 1} - 3}{18 x - 6}\right) \]
lhopitalApply l'Hôpital's rule once✓ Proved
\[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{e^{3 x - 1}}{2}\right) \]
lhopitalsimplifyApply l'Hôpital's rule again Simplify the constant factor✓ Proved
\[ = \frac{1}{2} \]
limitEvaluate the remaining 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
limitsimplifyThe product (2*x+1)**2*exp(-2*x) tends to 0 as x→∞ Multiply by exp(-1) gives 0✓ 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
rewritesimplifyUse the Taylor expansion exp(3*x)=1+3*x+9/2*x**2+… Cancel the common factor 9*x**2✓ Proved
\[ = \frac{1}{2} \]
limitThe limit of a constant is the constant itself✓ 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
rewriteUse the Taylor series expansion for cos(2x).✓ Proved
\[ = \lim_{x \to 0^+} \frac{1}{6} \]
simplifyalgebracancelKeep the leading term of the expansion. Expand the squared term. Cancel the x**2 terms.✓ Proved
\[ = \frac{1}{6} \]
simplifyFinal evaluation.✓ Proved
Answer \( \frac{1}{6} \)
The graph of f(x) = (2x - sin(2x))/(8x³). Two points slide along the curve toward x = 0, one from each side; their heights close in on 1/6.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to 0^+}\left(\frac{2 x - \sin{\left(2 x \right)}}{8 x^{3}}\right) \]
limit✓ Proved
\[ = \lim_{x \to 0^+} \frac{1}{6} \]
rewritesimplifycancelreplace sin(2*x) by its series 2*x - 8*x**3/6 subtract the 2*x terms cancel 8*x**3 in numerator and denominator✓ Proved
\[ = \frac{1}{6} \]
limitevaluate the constant 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
simplifysimplifyrewriteCompute the derivatives in the numerator and denominator. Simplify the fraction. Rewrite the expression using a positive exponent.✓ Proved
\[ = 0 \]
limitThe limit of exp(-2x) as x approaches infinity is 0.✓ Proved
Answer \( 0 \)
The graph of f(x) = 4x²·e^(-2x). A point runs right along the curve; its height settles toward 0.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
limitAs x→∞, exp(2*x) dominates, so the limit is 0.✓ 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
limitSince exp(1) is a constant, it can be omitted from the limit.✓ Proved
\[ = 0 \]
limitThe exponential term exp(-3*x) decays faster than any polynomial, so the product tends to 0.✓ 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 to (x+1)**2/exp(x).✓ Proved
\[ = \lim_{x \to \infty}\left(2 e^{- x}\right) \]
lhopitalApply l'Hôpital again to 2*(x+1)/exp(x).✓ Proved
\[ = 0 \]
limitAs x→∞, 2/exp(x)→0.✓ 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
factorrewriteFactor out the constant exp(1). Rewrite exp(-x) as 1/exp(x).✓ Proved
\[ = e \lim_{x \to \infty}\left(\left(2 x - 2\right) e^{- x}\right) \]
lhopitalApply l'Hôpital's rule once: derivative of numerator 2*(x-1), derivative of denominator exp(x).✓ Proved
\[ = e \lim_{x \to \infty}\left(2 e^{- x}\right) \]
lhopitalApply l'Hôpital's rule again: derivative of numerator 2, derivative of denominator exp(x).✓ Proved
\[ = 0 \]
limitsimplifyAs x→∞, exp(x)→∞, so 2/exp(x)→0. Multiply by exp(1) gives 0.✓ 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
\[ = \lim_{x \to 0^+}\left(\frac{2 e^{4 x}}{\cos{\left(2 x \right)}}\right) \]
simplifysimplifylimitCompute the derivatives in the numerator and denominator. Simplify the fraction by canceling the common factor of 2. The expression is now continuous at x = 0.✓ Proved
\[ = 2 \]
substitutionsimplifysimplifySubstitute x = 0 into the expression. Evaluate the exponential and trigonometric functions at 0. Final result.✓ Proved
Answer \( 2 \)
The graph of f(x) = (e^(4x) - 1)/sin(2x). Two points slide along the curve toward x = 0, one from each side; their heights close in on 2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to 0^+}\left(\frac{e^{4 x} - 1}{\sin{\left(2 x \right)}}\right) \]
\[ = \lim_{x \to 0^+}\left(\frac{2 e^{4 x}}{\cos{\left(2 x \right)}}\right) \]
simplifyCompute the derivatives✓ Proved
\[ = 2 \]
limitEvaluate the limit by substitution✓ 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
factorlimit-lawTake the constant 2 out of the limit. Reorder constants.✓ Proved
\[ = 0 \]
limitlimitx*exp(-2*x) -> 0 as x -> oo. 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 \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{e^{4 x - 2} - 1}{\sin{\left(2 x - 1 \right)}} \).
\[ \lim_{x \to \frac{1}{2}^+}\left(\frac{e^{4 x - 2} - 1}{\sin{\left(2 x - 1 \right)}}\right) \]
limitStart with the original limit.✓ Proved
\[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(e^{4 x - 2} - 1\right)}{\frac{d}{d x} \sin{\left(2 x - 1 \right)}}\right) \]
lhopitalApply L'Hopital's rule because the limit is of the form 0/0.✓ Proved
\[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{2 e^{4 x - 2}}{\cos{\left(2 x - 1 \right)}}\right) \]
simplifysimplifyCompute the derivatives. Simplify the fraction by dividing by 2.✓ Proved
\[ = \lim_{x \to \frac{1}{2}^+} 2 \]
limitsimplifysimplifyEvaluate the limit by substituting x = 1/2. Simplify the exponents and trigonometric arguments. Evaluate the exponential and cosine functions.✓ Proved
\[ = 2 \]
simplifyFinal result.✓ Proved
Answer \( 2 \)
The graph of f(x) = (e^(4x - 2) - 1)/sin(2x - 1). Two points slide along the curve toward x = 1/2, one from each side; their heights close in on 2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
\[ \lim_{x \to \frac{1}{2}^+}\left(\frac{e^{4 x - 2} - 1}{\sin{\left(2 x - 1 \right)}}\right) \]
limitrewriteRewrite using (x-1/2) to prepare for l'Hôpital.✓ Proved
\[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(e^{4 x - 2} - 1\right)}{\frac{d}{d x} \sin{\left(2 x - 1 \right)}}\right) \]
lhopital✓ Proved
\[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{2 e^{4 x - 2}}{\cos{\left(2 x - 1 \right)}}\right) \]
simplifysimplify✓ Proved
\[ = 2 \]
limitlimit✓ 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
\[ = 2 \]
limitalgebrasimplifysimplifyEvaluate the limit by substituting x = 1. Simplify the arguments of the exponential and trigonometric functions. Evaluate the exponential and trigonometric values. Final result.✓ Proved
Answer \( 2 \)
The graph of f(x) = (e^(2x - 2) - 1)/sin(x - 1). Two points slide along the curve toward x = 1, one from each side; their heights close in on 2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
limitThe first limit equals 1 because (exp(u)-1)/u -> 1 as u->0.✓ Proved
\[ = 2 \]
limitlimitThe second limit equals 2 because (x-1)/sin(x-1) -> 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
✓ 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
simplifysimplifyCompute the derivatives. Cancel the common factor of 3.✓ Proved
\[ = 0 \]
limitalgebrasimplifysimplifyEvaluate the limit by substituting x = 1/3. Simplify the expression inside the sine and exponential functions. Evaluate the trigonometric and exponential functions. Final result.✓ Proved
Answer \( 0 \)
The graph of f(x) = (cos(3x - 1) - 1)/(e^(3x - 1) - 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.
\[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\cos{\left(3 x - 1 \right)} - 1}{e^{3 x - 1} - 1}\right) \]
limitAs u -> 0, -sin(u) -> 0 and exp(u) -> 1, so the limit is 0.✓ 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
limit-lawApply the quotient rule for limits.✓ Proved
\[ = 2 \]
limitsimplifysimplifyEvaluate the limits of the numerator and denominator separately. Simplify the exponential and trigonometric terms. Final result.✓ Proved
Answer \( 2 \)
The graph of f(x) = (e^(2x) - 1)/sin(x). Two points slide along the curve toward x = 0, one from each side; their heights close in on 2.
A second solution, written separately
By gpt-oss:20b, without seeing the first. SymPy checked every line.
✓ 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