Question

Calculate the following limits \lim_{x\rightarrow a}\frac{x^{3}-a^{3}}{x^{4}-a^{4}}

Polynomial factorization
ANSWERED
asked 2021-06-08
Calculate the following limits
\(\lim_{x\rightarrow a}\frac{x^{3}-a^{3}}{x^{4}-a^{4}}\)

Answers (1)

2021-06-09
Step 1
Given
\(\lim_{x\rightarrow a}\frac{x^{3}-a^{3}}{x^{4}-a^{4}}\)
We have to evaluate the limit.
Step 2
We have
\(\lim_{x\rightarrow a}\frac{x^{3}-a^{3}}{x^{4}-a^{4}}\)
\(\Rightarrow \lim_{x\rightarrow a}\frac{(x-a)(x^{2}+xa+a^{2})}{(x^{2}-a^{2})(x^{2}+a^{2})}\) (Done factorization)
\(\Rightarrow \lim_{x\rightarrow a}\frac{(x-a)(x^{2}+xa+a^{2})}{(x+a)(x-a)(x^{2}+a^{2})}\) (Again done factorization)
\(\Rightarrow \lim_{x\rightarrow a}\frac{(x^{2}+xa+a^{2})}{(x+a)(x^{2}+a^{2})}\) (Cancelling out common factor)
\(\Rightarrow \lim_{x\rightarrow a}\frac{(a^{2}+a\times a+a^{2})}{(a+a)(a^{2}+a^{2})}\) (Substituting the limit)
\(\Rightarrow \lim_{x\rightarrow a}\frac{(3a^{2})}{(2a)(2a^{2})}\)
\(\Rightarrow \frac{3}{2\times (2a)}\)
\(\Rightarrow \frac{3}{4a}\)
So,
\(\lim_{x\rightarrow a}\frac{x^{3}-a^{3}}{x^{4}-a^{4}}=\frac{3}{4a}\)
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