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Chapter 1: Q. 53 (page 136)

$\lim_{x 鈫� 0} (\frac{2 x}{e^{x} - 1})$

Short Answer

Expert verified

$\lim_{x 鈫� 0} (\frac{2 x}{e^{x} - 1}) = 2$

Step by step solution

01

Step 1. Given information

$\lim_{x 鈫� 0} (\frac{2 x}{e^{x} - 1})$

02

Step 2.Simplify

Since,

\lim_{x 鈫� a} [c 路 f (x)] = c 路 \lim_{x 鈫� a} f (x)

Therefore,

\lim_{x 鈫� 0} (\frac{2 x}{e^{x} - 1}) = 2 路 \lim_{x 鈫� 鈥� 0} (\frac{x}{e^{x} - 1})

03

Step 3. Calculate the value

Apply L'Hopital's rule,

$\lim_{x 鈫� 0} (\frac{2 x}{e^{x} - 1}) = 2 路 \lim_{x 鈫� 鈥� 0} (\frac{1}{e^{x}})$

role="math" localid="1648011355231" $\lim_{x 鈫� 0} (\frac{2 x}{e^{x} - 1}) = 2 \frac{1}{e^{0}}$

$\lim_{x 鈫� 0} (\frac{2 x}{e^{x} - 1}) = 2$

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Most popular questions from this chapter

In Exercises 39鈥�44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

$f (x) = \{\begin{array}{l} (x - 3), i f x < 0 \\ - (x - 3), i f x 猢� 0 \end{array} .$

Write delta-epsilon proofs for each of the limit statements $\lim_{x 鈫� c} f (x) = L$ in Exercises $47 - 60$ .

$\lim_{x 鈫� - 6} (x + 2) = - 4$ .

Write delta-epsilon proofs for each of the limit statements $\lim_{x 鈫� c} f (x) = L$ in Exercises $47 - 60$ .

$\lim_{x 鈫� - 3} (1 - x) = 4$ .

Write delta-epsilon proofs for each of the limit statements $\lim_{x 鈫� c} f (x) = L$ in Exercises $47 - 60$ .

$\lim_{x 鈫� 3} (x^{2} - 6 x + 11) = 2$ .

If $f$ is a continuous function, what can you say about $\lim_{x 鈫� 1} f (x) ?$

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