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Jhevon · #2 October 1st, 2007, 10:03 PM

Quote:

Originally Posted by Amadeus

no one? T_T

be patient

Quote:

Originally Posted by Amadeus

Determine the limit of the trigonometric function (if it exists).

1. lim sin x / 5x
(x -> 0)
$lim sin x / 5x (x -> 0)$

Note that: $\lim_{x \to 0} \frac {\sin x}{5x} = \frac 15 \lim_{x \to 0} \frac {\sin x}x$

you should know what $\lim_{x \to 0} \frac {\sin x}x$ is, so continue.

Quote:

2. lim tan^2x / x
(x ->0)

multiply the top and bottom by x:

we get: $\lim_{x \to 0} \frac {x \tan^2 x}{x^2} = \lim_{x \to 0} x \left( \frac {\tan x}{x} \right)^2$

Now, $\lim_{x \to 0} \frac {\tan x}x = \lim_{x \to 0} \frac {\sin x}x$ , so you should be able to finish this one too

Quote:

3. lim cos x tan x / x
(x -> 0)

$\lim_{x \to 0} \cos x \left( \frac {\tan x}x\right) = \lim_{x \to 0}\cos x \cdot \lim_{x \to 0} \frac {\tan x}x$

same story as the last one. perhaps a nicer way to do it is to realize that $\tan x = \frac {\sin x}{\cos x}$ , so the limit would simplify to: $\lim_{x \to 0} \frac {\sin x}x$

...i guess you're starting to see how important the limit $\lim_{x \to 0} \frac {\sin x}x$ is, huh? ...you may want to simplify this way in the last problem

Quote:

Below, find lim f(x+h) - f(x) / h
----------- (x->0)

1. f(x) = radical x

You want: $\lim_{h \to 0} \frac {\sqrt{x + h} - \sqrt {x}}h$

rationalize the numerator and simplify