pedzenekO

2021-06-08

Calculate the following limits
$\underset{x\to a}{lim}\frac{{x}^{3}-{a}^{3}}{{x}^{4}-{a}^{4}}$

dieseisB

Step 1
Given
$\underset{x\to a}{lim}\frac{{x}^{3}-{a}^{3}}{{x}^{4}-{a}^{4}}$
We have to evaluate the limit.
Step 2
We have
$\underset{x\to a}{lim}\frac{{x}^{3}-{a}^{3}}{{x}^{4}-{a}^{4}}$
$⇒\underset{x\to a}{lim}\frac{\left(x-a\right)\left({x}^{2}+xa+{a}^{2}\right)}{\left({x}^{2}-{a}^{2}\right)\left({x}^{2}+{a}^{2}\right)}$ (Done factorization)
$⇒\underset{x\to a}{lim}\frac{\left(x-a\right)\left({x}^{2}+xa+{a}^{2}\right)}{\left(x+a\right)\left(x-a\right)\left({x}^{2}+{a}^{2}\right)}$ (Again done factorization)
$⇒\underset{x\to a}{lim}\frac{\left({x}^{2}+xa+{a}^{2}\right)}{\left(x+a\right)\left({x}^{2}+{a}^{2}\right)}$ (Cancelling out common factor)
$⇒\underset{x\to a}{lim}\frac{\left({a}^{2}+a×a+{a}^{2}\right)}{\left(a+a\right)\left({a}^{2}+{a}^{2}\right)}$ (Substituting the limit)
$⇒\underset{x\to a}{lim}\frac{\left(3{a}^{2}\right)}{\left(2a\right)\left(2{a}^{2}\right)}$
$⇒\frac{3}{2×\left(2a\right)}$
$⇒\frac{3}{4a}$
So,
$\underset{x\to a}{lim}\frac{{x}^{3}-{a}^{3}}{{x}^{4}-{a}^{4}}=\frac{3}{4a}$

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