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Evaluate $\lim _{x \rightarrow 3}\left(\frac{x^{4}-81}{x-3}\right)$



To evaluate: $\lim _{x \rightarrow 3} \frac{x^{4}-81}{x-3}$

Formula used:

We have,

$\lim _{x \rightarrow a} f(x)=f(a)$ and

As $\mathrm{x} \rightarrow 3$, we have

$\lim _{x \rightarrow 3} \frac{x^{4}-81}{x-3}=\lim _{x \rightarrow 3} \frac{\left(x^{2}+9\right)\left(x^{2}-9\right)}{x-3}$

$\lim _{x \rightarrow 3} \frac{x^{4}-81}{x-3}=\lim _{x \rightarrow 3} \frac{\left(x^{2}+9\right)(x+3)(x-3)}{x-3}$

$\lim _{x \rightarrow 3} \frac{x^{4}-81}{x-3}=\lim _{x \rightarrow 3}\left(x^{2}+9\right)(x+3)$

$\lim _{x \rightarrow 3} \frac{x^{4}-81}{x-3}=(9+9)(3+3)$

$\lim _{x \rightarrow 3} \frac{x^{4}-81}{x-3}=486$

Thus, the value of $\lim _{x \rightarrow 3} \frac{x^{4}-81}{x-3}$ is 486 .


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