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Chapter 3: Q. 24 (page 310)

Calculate each of the limits in Exercises 21鈥�48. Some of these limits are made easier by L鈥橦opital鈥檚 rule, and some are not
$\lim_{x 鈫� 鈭�} \frac{e^{x} - 7}{8 x^{2} + 12 x + 5}$

Short Answer

Expert verified

The value of the given limit $\lim_{x 鈫� 鈭�} \frac{e^{x} - 7}{8 x^{2} + 12 x + 5} = 鈭�$

Step by step solution

01

Step 1. Given information

The given limit $\lim_{x 鈫� 鈭�} \frac{e^{x} - 7}{8 x^{2} + 12 x + 5}$

02

Step 2. Using L’Hopital’s rule find the value of given limit

The limit using L鈥橦opitals rule is given below:

$\lim_{x 鈫� 鈭�} \frac{e^{x} - 7}{8 x^{2} + 12 x + 5} = \lim_{x 鈫� 鈭�} \frac{e^{x} - 0}{8.2 x + 12.1 + 0} = \lim_{x 鈫� 鈭�} \frac{e^{x}}{16 x + 12} = \lim_{x 鈫� 鈭�} \frac{e^{x}}{16.1 + 0} = \lim_{x 鈫� 鈭�} \frac{e^{x}}{12} = \frac{e^{鈭�}}{16} = 鈭�$

Therefore,the value of the given limit $= 鈭�$

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Most popular questions from this chapter

Use a sign chart for $f^{'}$ to determine the intervals on which each function $f$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$f (x) = l n (l n (x))$

Restate Rolle鈥檚 Theorem so that its conclusion has to do with tangent lines.

For each set of sign charts in Exercises 53鈥�62, sketch a possible graph of f.

Sketch careful, labeled graphs of each function f in Exercises 63鈥�82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f, f', a n d f'',$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f (x) = \frac{x^{2} - x}{x^{2} - 3 x + 2}$

Sketch careful, labeled graphs of each function f in Exercises 63鈥�82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f, f', a n d f'',$ and examine any relevant limits so that you can describe all key points and behaviors of f.

$f (x) = \frac{e^{x}}{x}$

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