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Chapter 1: Q. 51 (page 136)

$\lim_{x 鈫� 0} (\frac{e^{x} - 1}{e^{2 x} + 2 e^{x} - 3})$

Short Answer

Expert verified

$\lim_{x 鈫� 0} (\frac{e^{x} - 1}{e^{2 x} + 2 e^{x} - 3}) = \frac{1}{4}$

Step by step solution

Step 1. Given information

$\lim_{x 鈫� 0} (\frac{e^{x} - 1}{e^{2 x} + 2 e^{x} - 3})$

Step 2. Plug in the values

$\lim_{x 鈫� 0} (\frac{e^{x} - 1}{e^{2 x} + 2 e^{x} - 3}) = \lim_{x 鈫� 鈥� 0} (\frac{1}{e^{x} + 3})$

$\lim_{x 鈫� 0} (\frac{e^{x} - 1}{e^{2 x} + 2 e^{x} - 3}) = \frac{1}{e^{0} + 3}$

$\lim_{x 鈫� 0} (\frac{e^{x} - 1}{e^{2 x} + 2 e^{x} - 3}) = \frac{1}{4}$

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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.

$f (x) = \{\begin{matrix} x + 1, i f x < 1 \\ 3 x - 1, i f 1 猢� x < 2 \\ x + 2, i f x 猢� 2 \end{matrix} .$

Use what you know about one-sided limits to prove that a function $f$ is continuous at a point $x = c$ if and only if it is both left and right continuous at $x = c$ .

For each functionf graphed in Exercises23鈥�26, describe the intervals on whichf is continuous. For each discontinuity off, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements.

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

$\lim_{x 鈫� 1} \frac{x^{2} - 1}{x - 1} = 2$

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

$\lim_{x 鈫� 2} (3 x^{2} - 12 x + 15) = 3$ .

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limx鈫�0ex-1e2x+2ex-3

Short Answer

Step by step solution

Step 1. Given information

Step 2. Plug in the values

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$\lim_{x 鈫� 0} (\frac{e^{x} - 1}{e^{2 x} + 2 e^{x} - 3})$