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My teachers key says that it us -2. I was never taught how to evaluate limits that go to infinity or negative infinity so I need to be taught how to solve them.

My teachers key says that it us -2. I was never taught how to evaluate limits that-example-1

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SOLUTION

The given limit is:


\lim_(x\to-\infty)(√(4x^2+5))/(x-3)

Divide by the numerator and denominator by x


\operatorname{\lim}_(x\to-\infty)((√(4x^2+5))/(x))/(1-(3)/(x))

Upon simplifying this gives:


\begin{gathered} \frac{\lim_(x\to-\infty)\sqrt{(4x^2)/(x^2)-(5)/(x^2)}}{\lim_(n\to\infty)(1-(3)/(x))} \\ =\frac{\operatorname{\lim}_(x\to-\infty)\sqrt{4-(5)/(x^2)}}{\operatorname{\lim}_(n\to\infty)(1-(3)/(x))} \end{gathered}

Taking the limit gives:


\begin{gathered} \frac{\sqrt{4+(5)/(-\infty)}}{1-(3)/(-\infty)} \\ (√(4+0))/(1-0) \\ =(\pm2)/(1) \\ =\pm2 \end{gathered}

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