Question

Estimate the limit numerically or state that the limit does not exist, DNE:

lim(x-> 0) sin(65x)/(x)

Answer 1

By estimation, I assume you mean taking values close to zero.

What is the value of \(\dfrac{\sin 65 x}{x}\) when \(x=0.1\) ? when \(x=0.01\) ? when \(x=0.001\) ? And so on.

Similarly, you must check from the other direction. What is the value of \(\dfrac{\sin 65 x}{x}\) when \(x=-0.1\) ? when \(x=-0.01\) ? when \(x=-0.001\) ?

Alternatively, if you're familiar with the limit \(\displaystyle\lim_{x\to0}\frac{\sin ax}{ax}=1\), you can make some simple manipulations:
\[\lim_{x\to0}\frac{\sin65x}{x}=65\lim_{x\to0}\frac{\sin65x}{65x}=65\]

Answer 2

This helped, ty.

Answer 3

yw