Originally Posted by

**NBrunk** Hello, and thanks for the help in advance.

I have two limits I must evaluate on a practice exam, and am having a little trouble. They are both complicated enough that you cannot simply plug in the value the variable is approaching. They are listed below.

limit __sin(4x)__ As x -> 0

.........2x

$\displaystyle \color{red}\mbox{If you already know that } \frac{\sin x}{x} \xrightarrow [x \to 0] {} 1 \mbox{ , then it follows that } \frac{\sin f(x)}{f(x)} \xrightarrow [x \to 0] {} 1$

$\displaystyle \color{red}\mbox{for any function }f(x)\,\mbox{ s.t. }f(x) \xrightarrow [x \to 0] {} 0\,\,when\,\, x\rightarrow 0\,\,and\,\,\sin f(x)\,\,exists$

$\displaystyle \color{red}\mbox{Well, now just write }\frac{\sin 4x}{2x}=2\cdot \frac{\sin 4x}{4x}....$

limit __1 - cos(t)__ As t -> 0

.........t^2

$\displaystyle \color{red}1-\cos t=2\sin^2(t/2)\Longrightarrow\,\frac{1-\cos t}{t^2}=\frac{1}{2}\cdot\left(\frac{\sin \frac{t}{2}}{\frac{t}{2}}\right)^2 ...$

$\displaystyle \color{blue}Tonio$

(The periods are simply so that the fractions look right.)

I know you can pull 1/2 out in front of the limit for the first example (with x), but other than that I am at a loss as to what to do. I can do more complicated things...it's just been a while since I had to evaluate limits like this. Maybe it's a few simple things I'm forgetting?

Thanks again, I hope the way I've typed them is legible.