Evaluating $f(x)$ for values of $x$ that approach $0$

Question

(a) By graphing the function $$f(x) = \frac{\cos 2x − cos x}{x^2}$$ and zooming in toward the point where the graph crosses the y-axis, estimate the value of $\lim_{x \to 0} f(x)$.

I find that the answer here is $-1.5$ but I can't solve part (b)

(b) Check your answer in part (a) by evaluating $f(x)$ for values of x that approach $0$. (Round your answers to six decimal places.) $f(0.1)=?$ $f(00.1)=?$ $f(000.1)=?$ $f(0000.1)=?$ $f(−0.1)=?$ $f(−00.1)=?$ $f(−000.1)=?$ $f(−0000.1)=?$

Any help is greatly appreciated. I don't have a graphic calculator. Thanks.

Answer 1

Just use your scientific calculator to evaluate each value on your function. For example, for your first value, $f(0.1)=\frac{\cos(2 \times 0.1)-\cos(0.1)}{0.1^2}=\frac{0.980067-0.995004}{0.01}=-1.493700$. Do the same for each value and you'll see that they will be approaching to $0$ when you evaluate them on your function.

By the way, I think you have an error in your statement because $0.1=00.1=000.1$, I think you meant $0.1, 0.01, 0.001$, etc.

Answer 2

$$ {1 - 2x^{2} + 4x^{3}/3 - 1 + x^{2}/2 - x^{3}/6 \over x^{2}} \approx -\,{3 \over 2} + {7 \over 6}\, x\,, \qquad \left\vert x\right\vert \approx 0 $$