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Chapter 1: Q. 96 (page 137)

Use algebra, limit rules, and the continuity of $\ln x$ on $(0, 鈭�)$ to prove that every logarithmic function of the form $f (x) = \log_{b} x$ is continuous on $(0, 鈭�)$ .

Short Answer

Expert verified

It is proved that every logarithmic function of the form $f (x) = \log_{b} x$ is continuous on $(0, 鈭�)$ .

Step by step solution

01

Step 1. Given Information

We are given a function,

$f (x) = \log_{b} x$

02

Step 2. Proving the statement.

Given that $c 鈭� 鈩�$ ,

$\lim_{x 鈫� c} f (x) = \lim_{x 鈫� c} A b^{x} = A \lim_{x 鈫� c} b^{x} = A \lim_{x 鈫� c} {(e^{\ln b})}^{x} = A {(\lim_{x 鈫� c} e^{x})}^{\ln b} = A {(e^{c})}^{\ln b} = A {(e^{\ln b})}^{c} = A b^{c} = f (c)$

Hence, $f (x) = \log_{b} x$ is continuous on $(0, 鈭�)$ .

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Most popular questions from this chapter

Calculate each of the limits:

$\lim_{x 鈫� 3} \frac{\sqrt{x}}{\tan^{- 1} \sqrt{x}}$

For each limit statement , use algebra to find 未 > 0 in terms of $蔚$ > 0 so that if 0 < |x 鈭� c| < 未, then | f(x) 鈭� L| < $蔚$ .

$\lim_{x 鈫� 3} \frac{1}{x} = \frac{1}{3}; you may assume 未鈮� 1$

For each limit statement in Exercises $41 - 44$ , use algebra to find $未$ or $N$ in terms of $蔚$ or $M$ , according to the appropriate formal limit definition.

$\lim_{x 鈫� 鈭�} \frac{x - 1}{x} = 1$ , find $N$ in terms of $蔚$ .

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 a delta鈥揺psilon proof that proves that $f$ is continuous on its domain. In each case, you will need to assume that 未 is less than or equal to $1$ .

$f (x) = x^{鈭� 1 / 2}$

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