Question 1137967
Find lim( as theta approaches 0) 1-cos^2 theta/ (theta)^2 
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Use parentheses to make your question clear:

"lim( as theta approaches 0) 1-cos^2 theta/ (theta)^2 "  is  {{{-infinity}}}<br>

But you probably wanted this:
lim({{{theta}}}-->0) {{{ ((1-cos^2(theta))/theta^2) }}}<br>

= lim({{{theta}}}-->0) {{{ ((sin^2(theta))/theta^2) }}}<br>
This is 0/0 interderminate form, use L'Hospital's rule: lim(f(x)/g(x)) = lim(f'(x)/g'(x)):<br>
= lim({{{theta}}}-->0) {{{ ((2sin(theta)cos(theta))/(2*theta)) }}}<br>

Still 0/0, apply L'Hospital's again:
= lim({{{theta}}}-->0) {{{ ((cos^2(theta)-sin^2(theta))/(1)) }}}<br>
= 1/1 - 0/1 =  {{{ highlight( 1 ) }}}<br>

Here is a graph:

{{{ graph(400,400, -12,12,-1,2, sin^2(x)/x) }}}