Question-2

Evaluate the following limit:

\[ \lim_{x \rightarrow 2} \cfrac{x^6 -24x -16}{x^3 +2x - 12} \]

Hint

L’Hopital’s rule

Answer

\(12\)

Solution

This limit has \(\cfrac{0}{0}\) form. So we can try the L’Hopital’s rule. \[ \begin{aligned} \lim \limits_{x \rightarrow 2} \cfrac{x^6 -24x -16}{x^3 +2x - 12} &= \lim \limits_{x \rightarrow 2} \cfrac{6x^5 -24}{3x^2 + 2} = 12 \end{aligned} \] Note that the L’Hopital rule works here because the quotient of the derivatives of the numerator and denominator exists and is equal to \(12\). For all the conditions that have to be satisfied to apply the rule, check this Wikipedia entry.