SOLUTION: Find lim( as theta approaches 0) 1-cos^2 theta/ (theta)^2

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Question 1137967: Find lim( as theta approaches 0) 1-cos^2 theta/ (theta)^2
Found 2 solutions by MathLover1, math_helper:
Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!
lim%28%28theta%29-%3E0%29+%281+-+cos%5E2%28theta%29%29%2F%28theta%29%5E2+=

Apply L'Hopital's Rule

lim%28theta-%3E0%29+%28sin%5E2%282theta%29%29%2F2%28theta%29 =

Apply L'Hopital's Rule

lim%28theta-%3E0%29+%28cos%5E2%282theta%29%2A2%29%2F2 =

lim%28theta-%3E0%29 %28cos%5E2%282theta%29%29 =

Plug in the value : theta=0

lim%28theta-%3E0%29+%28cos%5E2%282%2A0%29%29 =

lim%28theta-%3E0%29+%28cos%5E2%280%29%29+=1



Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
Find lim( as theta approaches 0) 1-cos^2 theta/ (theta)^2
--------------
Use parentheses to make your question clear:
"lim( as theta approaches 0) 1-cos^2 theta/ (theta)^2 " is -infinity

But you probably wanted this:
lim(theta-->0) +%28%281-cos%5E2%28theta%29%29%2Ftheta%5E2%29+

= lim(theta-->0) +%28%28sin%5E2%28theta%29%29%2Ftheta%5E2%29+

This is 0/0 interderminate form, use L'Hospital's rule: lim(f(x)/g(x)) = lim(f'(x)/g'(x)):

= lim(theta-->0) +%28%282sin%28theta%29cos%28theta%29%29%2F%282%2Atheta%29%29+

Still 0/0, apply L'Hospital's again:
= lim(theta-->0) +%28%28cos%5E2%28theta%29-sin%5E2%28theta%29%29%2F%281%29%29+

= 1/1 - 0/1 = +highlight%28+1+%29+

Here is a graph:
+graph%28400%2C400%2C+-12%2C12%2C-1%2C2%2C+sin%5E2%28x%29%2Fx%29+