Evaluate the followingRute.5x²- 5w! 7L'Hopital's1-3X

Name: Evaluate the followingRute.5x²- 5w! 7L'Hopital's1-3X
Uploaded: 2024-03-20T06:46:03.000Z
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Description: Evaluate the followingRute.5x²- 5w! 7L'Hopital's1-3X

Answered: Ex w! 7 5x - 5 XS -

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter1: Introduction To Algebra

Section1.4: Translating Words Into Symbols

Problem 2MRE

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Question

Evaluate the following
Rute.
5x²- 5
w! 7
L'Hopital's
1-
3X

Expert Solution

Step 1

Evaluate the following limit.

Step 2

It is given that $\lim_{x \to \infty} \frac{5 x^{2} - 5}{e^{10 x}} + \lim_{x \to 0} \frac{1 - e^{- 5 x}}{e^{3 x}}$ .

Find the value of $\lim_{x \to \infty} \frac{5 x^{2} - 5}{e^{10 x}}$ .

$\lim_{x \to \infty} \frac{5 x^{2} - 5}{e^{10 x}} = \lim_{x \to \infty} \frac{5 x^{2}}{e^{10 x}} - \lim_{x \to \infty} \frac{5}{e^{10 x}}$

Apply L'Hopsital Rule for $\lim_{x \to \infty} \frac{5 x^{2}}{e^{10 x}}$ .

$\begin{array}{rcl} \lim_{x \to \infty} \frac{5 x^{2}}{e^{10 x}} & = & \lim_{x \to \infty} \frac{\frac{d}{d x} (5 x^{2})}{\frac{d}{d x} (e^{10 x})} \\ = & \lim_{x \to \infty} \frac{(10 x)}{(10 e^{10 x})} \\ = & \lim_{x \to \infty} \frac{\frac{d}{d x} (10 x)}{\frac{d}{d x} (10 e^{10 x})} \\ = & \lim_{x \to \infty} \frac{10}{100 e^{10 x}} \\ = & \frac{10}{100 e^{10 (\infty)}} \\ = & \frac{10}{\infty} \\ = & 0 \end{array}$

The limit $\lim_{x \to \infty} \frac{5 x^{2} - 5}{e^{10 x}} = \lim_{x \to \infty} \frac{5 x^{2}}{e^{10 x}} - \lim_{x \to \infty} \frac{5}{e^{10 x}}$ becomes as follows.

$\begin{array}{rcl} \lim_{x \to \infty} \frac{5 x^{2} - 5}{e^{10 x}} & = & \lim_{x \to \infty} \frac{5 x^{2}}{e^{10 x}} - \lim_{x \to \infty} \frac{5}{e^{10 x}} \\ = & 0 - \frac{5}{e^{10 (\infty)}} \\ = & 0 - 0 \\ = & 0 \end{array}$

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