limits - Find $lim_{xrightarrow -infty } frac{3x^{2}-sin(5x)}{x^{2}+2}$

$\lim_{x\rightarrow -\infty } \frac{3x^{2}-\sin(5x)}{x^{2}+2}$

A. 0

B. 1

C. 2

D. 3

After using L'hopital's rule again and again I got this expression:

$$\lim_{x\rightarrow -\infty } \frac{6+25\sin(5x)}{2}$$

But how do we proceed, what is the value of $\sin(-\infty)$?

Any help will be appreciated!

Answer

Application of L'hopitals rule for the first time gives,

$$\lim_{x \to -\infty} \frac{6x-5\cos (5x)}{2x}$$

We can try to do it again but it won't be useful $\lim_{x \to -\infty} \sin (x)$ does not exist because it oscillates between $-1$ and $1$ for $\frac{\pi}{2}+2\pi k$ and $\frac{3\pi}{2}+2\pi k$ so L'hopitals rule will not give a sensible answer.

Instead we may try more elementary approaches and write is as follows.

$$\lim_{x \to -\infty} \frac{6-5\frac{\cos(5x)}{x}}{2}$$

At this point note that the quaintly $\frac{\cos (5x)}{x}$ goes to zero because the top is bounded by $-1$ and $1$.

So the answer is $3$.