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Chapter 1: Q. 57 (page 149)

Calculate each limit in Exercises 35–80.
$\underset{x \to \infty}{l i m} \frac{4^{x} - 3^{x}}{5^{x}}$

Short Answer

Expert verified

The limit is 0

Step by step solution

01

Step 1. Given information

The given expression is $\underset{x \to \infty}{l i m} \frac{4^{x} - 3^{x}}{5^{x}}$

02

Step 2. Calculation

The limit is calculated as below,

$\underset{x \to \infty}{l i m} \frac{4^{x} - 3^{x}}{5^{x}} = \underset{x \to \infty}{l i m} (\frac{4^{x} - 3^{x}}{5^{x}} \times \frac{\frac{1}{5^{x}}}{\frac{1}{5^{x}}}) = \underset{x \to \infty}{l i m} (\frac{{(\frac{4}{5})}^{x} - {(\frac{3}{5})}^{x}}{1}) = (\frac{{(\frac{4}{5})}^{\infty} - {(\frac{3}{5})}^{\infty}}{1}) = \frac{0 - 0}{1} = 0$

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Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

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