Q. 95 Use algebra, limit rules, and th... [FREE SOLUTION]

Chapter 1: Q. 95 (page 137)

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

Short Answer

Expert verified

Ans: $\lim_{x 鈫� c} A b^{x} = f (c)$

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 $f (x) = A b^{x}$ is continuous everywhere.

Step 2. Prove:

$G i v e n c 鈭� 鈩�, \underset{x 鈫� c}{l i m} f (x) = \underset{x 鈫� c}{l i m} A b^{x} = \underset{x 鈫� c}{l i m} A \underset{x 鈫� c}{l i m} b^{x} = A {(\underset{x 鈫� c}{l i m} e^{l n (b)})}^{x} = A {(\underset{x 鈫� c}{l i m} e^{x})}^{l n (b)} = A {(e^{c})}^{l n (b)} = A {(e^{l n (b)})}^{c} = A b^{c} = f (c)$

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

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Prove:

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Use algebra, limit rules, and the continuity of exto prove that every exponential function of the form f(x)=Abxis continuous everywhere.

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Prove:

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