What is the limit as x approaches 0 of [(tanx)^2]/x?
lim tan^2/x= lim sinx/x *lim 1/cosx=1*1
see error corrected on other post.
To find the limit of the expression [(tanx)^2]/x as x approaches 0, we can use algebraic manipulation and trigonometric identities. Let's break down the steps:
1. Start with the expression [(tanx)^2]/x.
2. Rewrite the expression using the identity tan^2(x) = sin^2(x) / cos^2(x).
The expression becomes [sin^2(x) / cos^2(x)] / x.
3. Rearrange the expression by multiplying through by cos^2(x) to eliminate the denominator:
sin^2(x) / (cos^2(x) * x).
4. Consider the individual limits of sin^2(x) and cos^2(x) as x approaches 0.
The limit of sin^2(x) as x approaches 0 is 0, and the limit of cos^2(x) as x approaches 0 is 1.
5. Substitute the limits into the expression:
0/1 * 1/x = 0/x = 0.
6. Therefore, the limit of [(tanx)^2]/x as x approaches 0 is 0.
In summary, by rewriting the expression, applying trigonometric identities, and evaluating the limits of the individual terms, we determine that the limit is 0.