Find Limits of Trig Functions

G

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What are the steps to solve questions like the two below?

1. Find the limit of (1+cosx)/(x) as x-->0.

2. Find the limit of (1-cosx)/(x^2) as x-->0.

Do I simply plug 0 for x in BOTH questions and simplify?

Thanks,
Interval
 
For the 2nd one, since you have an indeterminate form,you could try L'Hopital's rule...twice.

\(\displaystyle \L\\\lim_{x\to\0}\frac{1-cos(x)}{x^{2}}\)

Differentiate top and bottom:

\(\displaystyle \L\\\lim_{x\to\0}\frac{sin(x)}{2x}\)

Again:

\(\displaystyle \L\\\lim_{x\to\0}\frac{cos(x)}{2}\)

Now, what's the limit?.
 
okay

Okay but our teacher is not up to L'Hopital Rule yet. Is there another way to find the limit?

We also have not covered differentiation of any kind.
 
Before going into L'Hopital, whenever we did limits my teacher would make us test increasingly close numbers to the number you're trying to find (.01, .00001 for limit x-->0)

it's a pain in the butt, but it's also another method.
 
Plug in x=0 and see what you get:

\(\displaystyle \L\\\frac{1-cos(x)}{x^{2}}\)


:roll:
 
For these, I believe the following theorems apply:

lim as x->0 of sinx / x = 1

lim as x->0 (1-cosx) /x = 0
 
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