8.  Evaluate the following limits   \lim_{x \rightarrow 3} \frac{x ^4 -81}{2x^2 -5x -3}

Answers (1)

The limit

\lim_{x \rightarrow 3} \frac{x ^4 -81}{2x^2 -5x -3}

At x = 2 both numerator and denominator becomes zero, so lets factorise the function

\lim_{x \rightarrow 3} \frac{(x-3)(x+3)(x^2+9)}{(x-3)(2x+1)}

\lim_{x \rightarrow 3} \frac{(x+3)(x^2+9)}{(2x+1)}

Now we can put the limit directly, so

\lim_{x \rightarrow 3} \frac{(x+3)(x^2+9)}{(2x+1)}

\Rightarrow \frac{((3)+3)((3)^2+9)}{(2(3)+1)}

\Rightarrow \frac{6\times18}{7}

\Rightarrow \frac{108}{7}

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