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Limits using Taylor series practice problems

Evaluate a 0/0 limit by replacing the numerator with its Taylor polynomial. 42 problems with worked solutions; in 42 of them every equation is proved by a computer algebra system.

4 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 7.599

medium

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 - 3 \cos{\left(x \right)}}{x^{2}} \).

Near 0, cos(x) - 1 = x**4/24 - x**2/2 + ⋯ (its Maclaurin series).
Reviewed
\[ \lim_{x \to 0}\left(\frac{- \frac{x^{4}}{24} + \frac{x^{2}}{2} + \cos{\left(x \right)} - 1}{x^{4}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{- \frac{x^{4}}{8} + \frac{3 x^{2}}{2}}{x^{2}} = \frac{3}{2} - \frac{x^{2}}{8} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{3 - 3 \cos{\left(x \right)}}{x^{2}}\right) = \frac{3}{2} \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( \frac{3}{2} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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

Problem 7.602

medium

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 x - 2 \operatorname{atan}{\left(x \right)}}{x^{3}} \).

Near 0, x - atan(x) = -x**5/5 + x**3/3 + ⋯ (its Maclaurin series).
Reviewed
\[ \lim_{x \to 0}\left(\frac{\frac{x^{5}}{5} - \frac{x^{3}}{3} + x - \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{- \frac{2 x^{5}}{5} + \frac{2 x^{3}}{3}}{x^{3}} = \frac{2}{3} - \frac{2 x^{2}}{5} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{2 x - 2 \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = \frac{2}{3} \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( \frac{2}{3} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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

Problem 7.603

medium

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{1 - \cos{\left(x \right)}}{x^{2}} \).

Near 0, cos(x) - 1 = x**4/24 - x**2/2 + ⋯ (its Maclaurin series).
Reviewed
\[ \lim_{x \to 0}\left(\frac{- \frac{x^{4}}{24} + \frac{x^{2}}{2} + \cos{\left(x \right)} - 1}{x^{4}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{- \frac{x^{4}}{24} + \frac{x^{2}}{2}}{x^{2}} = \frac{1}{2} - \frac{x^{2}}{24} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{1 - \cos{\left(x \right)}}{x^{2}}\right) = \frac{1}{2} \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( \frac{1}{2} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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

Problem 7.698

medium

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- 3 x + 3 \sin{\left(x \right)}}{x^{3}} \).

Near 0, -x + sin(x) = x**5/120 - x**3/6 + ⋯ (its Maclaurin series).
Reviewed
\[ \lim_{x \to 0}\left(\frac{- \frac{x^{5}}{120} + \frac{x^{3}}{6} - x + \sin{\left(x \right)}}{x^{5}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{\frac{x^{5}}{40} - \frac{x^{3}}{2}}{x^{3}} = \frac{x^{2}}{40} - \frac{1}{2} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{- 3 x + 3 \sin{\left(x \right)}}{x^{3}}\right) = - \frac{1}{2} \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( - \frac{1}{2} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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

More limits using taylor series practice

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- 3 x + 3 e^{x} - 3}{x^{2}} \).

Problem 7.600medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{\cos{\left(x \right)} - 1}{x \sin{\left(x \right)}} \).

Problem 7.601medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 \cos{\left(x \right)} - 2}{x^{2}} \).

Problem 7.604medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 x - 2 e^{x} + 2}{x^{2}} \).

Problem 7.605medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{\cos{\left(x \right)} - 1}{x^{2}} \).

Problem 7.606medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- x + \ln{\left(x + 1 \right)}}{x^{2}} \).

Problem 7.607medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 - 2 \cos{\left(x \right)}}{x \sin{\left(x \right)}} \).

Problem 7.608medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{x - e^{x} + 1}{x^{2}} \).

Problem 7.609medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{e^{x^{2}} - 1}{x^{2}} \).

Problem 7.610medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 x - 3 \ln{\left(x + 1 \right)}}{x^{2}} \).

Problem 7.611medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- 3 x + 3 \operatorname{atan}{\left(x \right)}}{x^{3}} \).

Problem 7.612medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 - 2 \cos{\left(x \right)}}{x^{2}} \).

Problem 7.613medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{1 - e^{x^{2}}}{x^{2}} \).

Problem 7.669medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- 2 x + 2 \sin{\left(x \right)}}{x^{3}} \).

Problem 7.670medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 e^{x^{2}} - 3}{x^{2}} \).

Problem 7.671medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 - 2 e^{x^{2}}}{x^{2}} \).

Problem 7.672medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 x - 2 \ln{\left(x + 1 \right)}}{x^{2}} \).

Problem 7.673medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{x - \operatorname{atan}{\left(x \right)}}{x^{3}} \).

Problem 7.674medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{1 - \cos{\left(x \right)}}{x \sin{\left(x \right)}} \).

Problem 7.675medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 x - 2 \sin{\left(x \right)}}{x^{3}} \).

Problem 7.676medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 \cos{\left(x \right)} - 2}{x \sin{\left(x \right)}} \).

Problem 7.677medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 \cos{\left(x \right)} - 3}{x^{2}} \).

Problem 7.678medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 x - 3 \sin{\left(x \right)}}{x^{3}} \).

Problem 7.679medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- 2 x + 2 \operatorname{atan}{\left(x \right)}}{x^{3}} \).

Problem 7.680medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 - 3 e^{x^{2}}}{x^{2}} \).

Problem 7.681medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{x - \sin{\left(x \right)}}{x^{3}} \).

Problem 7.682medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 e^{x^{2}} - 2}{x^{2}} \).

Problem 7.683medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- x + \sin{\left(x \right)}}{x^{3}} \).

Problem 7.699medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 \cos{\left(x \right)} - 3}{x \sin{\left(x \right)}} \).

Problem 7.700medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- x + e^{x} - 1}{x^{2}} \).

Problem 7.701medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 x - 3 \operatorname{atan}{\left(x \right)}}{x^{3}} \).

Problem 7.702medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- 2 x + 2 e^{x} - 2}{x^{2}} \).

Problem 7.703medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- x + \operatorname{atan}{\left(x \right)}}{x^{3}} \).

Problem 7.704medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- 3 x + 3 \ln{\left(x + 1 \right)}}{x^{2}} \).

Problem 7.705medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 - 3 \cos{\left(x \right)}}{x \sin{\left(x \right)}} \).

Problem 7.706medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 x - 3 e^{x} + 3}{x^{2}} \).

Problem 7.707medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{x - \ln{\left(x + 1 \right)}}{x^{2}} \).

Problem 7.708medium✓ Every equation proved

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- 2 x + 2 \ln{\left(x + 1 \right)}}{x^{2}} \).

Problem 7.709medium✓ Every equation proved