   Chapter 2.5, Problem 52E

Chapter
Section
Textbook Problem

In Exercises 45-56, find the values of x for which each function is continuous.52. f ( x ) = x − 1 x 2 + 2 x − 3

To determine
The value of x at which the function is continuous.

Explanation

Given information:

The function is,

f(x)=x1x2+2x3 (1)

Definition used:

A function f is continuous at x=a when it follows the following three conditions,

(1) f(a) is defined.

(2) limxaf(x) exists.

(3) limxaf(x)=f(a) .

Calculation:

For the function to be continuous, it should satisfy all the three conditions.

The given function is a rational function and the denominator x2+2x3 of the given function is 0 at x=1 and x=3 .

Factorize the denominator of equation (1),

f(x)=x1x2+2x3=(x1)(x1)(x+3) (2)

Substitute 1 for x in equation (2) to find the continuity of the function,

f(x)=(x1)(x1)(x+3)=11(11)(

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