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Chapter 4: Problem 26

Evaluate the following limits. $$\lim _{x \rightarrow 0} \frac{e^{x}-x-1}{5 x^{2}}$$

Short Answer

Expert verified
Question: Determine the limit of the given function as x approaches 0: $$\lim _{x \rightarrow 0} \frac{e^{x}-x-1}{5x^2}$$ Answer: The limit of the given function as x approaches 0 is \(\frac{1}{10}\).

Step by step solution

01

Rewrite the given limit

Rewrite the given limit as: $$\lim _{x \rightarrow 0} \frac{e^{x}-x-1}{5 x^{2}}$$
02

Apply L'H么pital's Rule

Since the limit is of indeterminate form, apply L'H么pital's Rule, which states that if the limit of the ratio of the derivatives of the functions exists, then the limit of the original functions exists as well and is equal to the limit of the ratio of the derivatives. By applying L'H么pital's Rule, we need to find the derivatives of the numerator and denominator: Numerator: \(\frac{d}{dx}(e^{x}-x-1) = e^{x}-1\) Denominator: \(\frac{d}{dx}(5x^2) = 10x\) Now rewrite the limit with these new expressions: $$\lim _{x \rightarrow 0} \frac{e^{x}-1}{10x}$$
03

Apply L'H么pital's Rule again

Since the new limit is still indeterminate, we must apply L'H么pital's Rule again. Numerator: \(\frac{d}{dx}(e^{x}-1) = e^{x}\) Denominator: \(\frac{d}{dx}(10x) = 10\) Now rewrite the limit again with these new expressions: $$\lim _{x \rightarrow 0} \frac{e^{x}}{10}$$
04

Evaluate the limit

The limit is no longer indeterminate, so we can directly substitute x = 0: $$\lim _{x \rightarrow 0} \frac{e^{x}}{10} = \frac{e^{0}}{10} = \frac{1}{10}$$ Thus, the limit of the given function as x approaches 0 is \(\frac{1}{10}\).

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