Ask question
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,
$$\lim_{x\rightarrow a}\frac{x^{3}-a^{3}}{x^{4}-a^{4}}=\frac{3}{4a}$$

expert advice

...