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  • Evaluate  <img alt="\lim _{x \rightarrow 2} \frac{x^4-16}{x^3-8}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Clim%20_%7Bx%20%5Crightarrow%202%7D%20%5Cfrac%7Bx%5E4-16%7D%7Bx%5E

Evaluate  \lim _{x \rightarrow 2} \frac{x^4-16}{x^3-8}

Option: 1

2


Option: 2

\frac{8}{3}


Option: 3

\frac{5}{3}


Option: 4

1


Answers (1)

best_answer

Given  \lim _{x \rightarrow 2} \frac{x^4-16}{x^3-8}

When applied limits it assumes the indeterminate form \frac{0}{0}

So, divide the numerator and denominator with x-2

\begin{aligned} \lim _{x \rightarrow 2} \frac{x^4-16}{x^3-8} & =\lim _{x \rightarrow 2} \frac{\left(\frac{x^4-16}{x-2}\right)}{\left(\frac{x^3-8}{x-2}\right)} \\ & =\frac{\lim _{x \rightarrow 2}\left(\frac{x^4-16}{x-2}\right)}{\lim _{x \rightarrow 2}\left(\frac{x^3-8}{x-2}\right)} \\ & =\frac{\lim _{x \rightarrow 2}\left(\frac{x^4-2^4}{x-2}\right)}{\lim _{x \rightarrow 2}\left(\frac{x^3-2^3}{x-2}\right)} \\ \end{aligned}

                          =\frac{4\left(2^{4-1}\right)}{3\left(2^{3-1}\right)} \\

                          =\frac{4\left(2^3\right)}{3\left(2^2\right)} \\

                         =\frac{8}{3}

Posted by

Rishabh

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