Question

How do you find `lim_(theta->0) tantheta/theta` using l'Hospital's Rule?

Answer 1

The derivative of `tan theta` is `sec^2 theta` and the derivative of `theta` is `1`.

`lim_(thetararr0)tantheta/theta = lim_(thetararr0)sec^2theta/1 = 1^2/1 = 1`

Answer 2

Determine that the expression evaluated at the limit results in an indeterminate form.
Differentiate the number and denominator.
The limit of the new expression is the same as the original.

Explanation:

The expression evaluated at `theta = 0` is the indeterminate form `0/0`, therefore we differentiate the numerator and denominator:

`lim_(thetararr0)((d(tan(theta)))/(d theta))/((d(theta))/(d theta))`

`lim_(thetararr0)sec^2(theta)/1 = sec^2(0) = 1`

Therefore, the limit of the original expression is same:

`lim_(thetararr0) (tan(theta))/theta = 1`