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

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:

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

Pure Maths

Geometry

Decision Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Statistics

Pure Maths

Geometry

Decision Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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.