Chapter 1: Q. 94 (page 137)

Use algebra, limit rules, and the continuity of $e^{x}$ to prove that every exponential function of the form $f (x) = A e^{kx}$ is continuous everywhere.

Short Answer

Expert verified

$T h e f u n c t i o n f (x) = A e^{kc} i s c o n t i n u o u s e v e r y w h e r e$ .

Step by step solution

Step 1. Given Information:

The strategy is to prove using algebra, limit rules, and the continuity of $e^{x}$ that every exponential function of the form $f (x) = A e^{k x}$ is continuous everywhere.

Step 2. Prove:

$T a k e l i m i t i n t h e e x p r e s s i o n e^{x} a s : \underset{x \to c}{l i m} e^{x} = e^{c} T h e r e f o r e, \underset{x \to c}{l i m} A e^{k x} = \underset{x \to c}{l i m} A \underset{x \to c}{l i m} e^{k x} = A \lim_{x \to c} e^{k x} = A e^{k (c)} = A e^{k c} H e n c e, t h e f u n c t i o n f (x) = A e^{kc} i s c o n t i n u o u s e v e r y w h e r e .$

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Use algebra, limit rules, and the continuity of $e^{x}$ to prove that every exponential function of the form $f (x) = A e^{kx}$ is continuous everywhere.

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Prove:

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Step by step solution

Step 1. Given Information:

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Use algebra, limit rules, and the continuity of $e^{x}$ to prove that every exponential function of the form $f (x) = A e^{kx}$ is continuous everywhere.