Answered: lim (e + x)/, for a constant a

Math

Calculus

lim (e + x)/, for a constant a

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter1: Equations And Inequalities

Section1.CT: Chapter Test

Problem 5CT: Solve for x: z=x.

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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

Evaluate the following limit. Use l’Hôpital’s Rule when it is convenient and applicable.

Expert Solution

Step 1

To evaluate the limit: $\lim_{x \to 0} {(e^{a x} + x)}^{\frac{1}{x}}$

Solution:

Given limit is of the form $1^{\infty}$ .

We know that if $\lim_{x \to a} {[f (x)]}^{g (x)}$ is $1^{\infty}$ form then:

$\lim_{x \to a} {[f (x)]}^{g (x)} = \lim_{x \to a} [f (x) - 1] \times g (x)$

Using the above property,

$\lim_{x \to 0} {(e^{a x} + x)}^{\frac{1}{x}} = \lim_{x \to 0} (e^{a x} + x - 1) \times \frac{1}{x}$

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