Define f(1) in a way that extends f(x) = {x^3 – 1} / {1 – x^2} to be continuous at x = 1. | Homework.Study.com

Question:

Define {eq}f(1)

{/eq} in a way that extends {eq}f(x)=\frac{x^3-1}{1-x^2}

{/eq} to be continuous at {eq}x=1

{/eq}.

The Concept of Removable Discontinuity

{eq}{/eq}

If a function f(x) has equal values of left hand limit and right hand limit for some x = a in its domain, but is undefined at that point, then function is said to have a removable discontinuity at x = a. Let,

{eq}\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = b \\

{/eq}

Then, we can define f(a) = b to remove the discontinuity at x = a. We will make use of this information to solve the given problem.

{eq}{/eq}

Answer and Explanation:

1

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{eq}{/eq}

Given function :

{eq}\displaystyle{ f(x)=\frac{x^3-1}{1-x^2} \\

\Rightarrow f(x) = \frac{(x-1)(x^2 + x +1)}{(1-x)(1+x)} \\ }

{/eq}

As…

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