The value of lim x → 1 e x 3 − x .

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 2, Problem 3RE
To determine

To find: The value of limx→1ex3−x.

Expert Solution

The limit of the function is 1.

Explanation of Solution

Definition 1: “A function f is continuous at a number a if limxaf(x)=f(a)”.

Definition 2: “If f is continuous at b and limxag(x)=b, then limxaf(g(x))=f(b). That is, limxaf(g(x))=f(limxag(x))”.

Calculation:

Obtain the limit of the function by using the definition 2.

Let h(x)=ex3x.

The given function h(x)=f(g(x)) is a composition of two functions namely, f(x)=ex and g(x)=x3x

The exponential function f(x)=ex and the polynomial function g(x)=x3x is continuous everywhere in the domain.

By Definition 1, consider limxag(x)=g(a) for a continuous functions.

limx1(x3x)=(131)=11=0

Therefore, limx1(x3x)=0.

By Definition 2, limxaf(g(x))=f(limxag(x)).

limx1ex3x=elimx1(x3x)=e0=1

Thus, the limit of the function is 1.

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