Q. 95

Question

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.

Step-by-Step Solution

Verified

Answer

Ans: $\lim_{x \to c} A b^{x} = f (c)$

1Step 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.

2Step 2. Prove:

$G i v e n c \in ℝ, \underset{x \to c}{l i m} f (x) = \underset{x \to c}{l i m} A b^{x} = \underset{x \to c}{l i m} A \underset{x \to c}{l i m} b^{x} = A {(\underset{x \to c}{l i m} e^{l n (b)})}^{x} = A {(\underset{x \to c}{l i m} e^{x})}^{l n (b)} = A {(e^{c})}^{l n (b)} = A {(e^{l n (b)})}^{c} = A b^{c} = f (c)$

Q. 94

Q. 96

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