what is the limit as x approaches a number if it goes to infinity from one end and negative infinity from the other? for example sec^2(x)/tanx as x approaches pi over two
-
sec(x)^2 / tan(x) =>
(1 + tan(x)^2) / tan(x) =>
1/tan(x) + tan(x)^2 / tan(x) =>
cot(x) + tan(x)
x goes to pi/2-
0 + inf
x goes to pi/2+
0 - inf
So, since the limits from either side approach different values (and since we're undefined at pi/2), then there is no limit
(1 + tan(x)^2) / tan(x) =>
1/tan(x) + tan(x)^2 / tan(x) =>
cot(x) + tan(x)
x goes to pi/2-
0 + inf
x goes to pi/2+
0 - inf
So, since the limits from either side approach different values (and since we're undefined at pi/2), then there is no limit
-
If the limit from the left is not equal to the limit from the right, then the limit does not exist.