Lim (tan^2(x)/x)
x->0
See:
http://www.jiskha.com/display.cgi?id=1283133068
To find the limit of the expression (tan^2(x)/x) as x approaches 0, we can use algebraic manipulation and several well-known limits.
Let's break down the expression and analyze it step by step:
lim (tan^2(x)/x) as x approaches 0.
First, let's simplify the expression. Recall the trigonometric identity: tan^2(x) = sec^2(x) - 1.
lim ((sec^2(x) - 1)/x) as x approaches 0.
Next, we'll divide both the numerator and denominator by x:
lim (sec^2(x)/x - 1/x) as x approaches 0.
Now, we have two terms in the expression. Let's analyze them separately.
The limit of (sec^2(x)/x) as x approaches 0 can be found by recognizing that sec^2(x) is the derivative of tan(x). Therefore, using a basic limit rule, we have:
lim (sec^2(x)/x) = lim (d/dx(tan(x))) as x approaches 0.
Differentiating tan(x) gives us sec^2(x), so:
lim (sec^2(x)/x) = lim (sec^2(x)) as x approaches 0.
Now, let's calculate the limit of sec^2(x) as x approaches 0. The standard result is:
lim (sec^2(x)) as x approaches 0 = 1.
For the second term, the limit of 1/x as x approaches 0 is:
lim (1/x) as x approaches 0 = ∞ (infinity).
Finally, we can combine the results:
lim ((sec^2(x)/x) - (1/x)) as x approaches 0 = 1 - ∞.
The expression 1 - ∞ is an indeterminate form. It does not give us a definitive answer. Therefore, the limit of (tan^2(x)/x) as x approaches 0 does not exist.