# evaluating limits of trig functions

• Aug 18th 2009, 12:39 PM
skeske1234
evaluating limits of trig functions
Evalue each of the following limits.

Ok I am having trouble with evaluating trig limits with the tan function. I was wondering if someone could show me step by step their answer for the following three questions:

a) lim (tanx)/(4x)
x-> 0

b) lim (x^3)/(tan^3(2x))
x->0

c) lim (2tanx)/(xsecx)
x->0

please explain it thoroughly as you can. Appreciate it very much. Thanks.
• Aug 18th 2009, 12:53 PM
Random Variable
a) $\displaystyle \lim_{x \to 0} \ \frac{\tan x}{4x} = \frac{1}{4} \lim_{x \to 0} \ \frac{\sin x}{x \cos x} = \frac{1}{4} \Big( \lim_{x \to 0} \ \frac{1}{\cos x} \cdot \lim_{x \to 0} \ \frac{\sin x}{x}\Big) = \frac{1}{4} \Big(1 \cdot 1\Big) = \frac{1}{4}$
• Aug 18th 2009, 01:04 PM
skeske1234
Quote:

Originally Posted by Random Variable
a) $\displaystyle \lim_{x \to 0} \ \frac{\tan x}{4x} = \frac{1}{4} \lim_{x \to 0} \ \frac{\sin x}{x \cos x} = \frac{1}{4} \Big( \lim_{x \to 0} \ \frac{1}{\cos x} \cdot \lim_{x \to 0} \ \frac{\sin x}{x}\Big) = \frac{1}{4} \Big(1 \cdot 1\Big) = \frac{1}{4}$

question.. when you break the eqtn into two, shown by the two limits at
1/cosx and sinx/x.. this step: isn't (1 x 1) wrong because sin0/0 is still undefined?
sorry, i'm not understanding that part
• Aug 18th 2009, 01:26 PM
skeske1234
actually I have it now, question 1 anyways.

For question two, this is what I have so far:

lim x^3/ [tan^(3) 2x]
x->0
= x^3 [ cos^(3) x/ sin^(3) x]
not sure what move to make next