# Math Help - Limits and Deivatives

1. ## Limits and Deivatives

I need help on the following questions. Please help if you can - I'm stuck.

Evaluate the limit (if it exists) as x goes to -6
lim x app -6 6 + 25x + 4x^2/(-36 + 6x + 2x^2)
The answer has to be in fraction form I'm told

Differentiate the following function with respect to x
g(x) = - 3/2 + 6/(5x^12/5) + 2x^9

Differentiate the following function with respect to x
f(x) = 9x^4 + (13x^10)/6

2. #1) So we have the limit: $\lim_{x \rightarrow -6} \frac{4x^2+25x+6}{2x^2+6x-36}$. Direct substitution gives 0/0. Uh oh. This is a perfect time to use L'Hopital's Rule, which for this case means that since the limit gave an indeterminate form, or 0/0, then the limit is actually equal to the derivative of the numerator divided by the derivative of the denominator. Thus the original limit is equal to $\frac{ \frac{d}{dx}(4x^2+25x+6)}{ \frac{d}{dx}(2x^2+6x-36)}$. Now try re-evaluating your limit.

3. Or if you don't know L'Hopital's rule yet, try factoring as $\frac{4x^2+25x+6}{2x^2+6x-36} = \frac{(4x+1)(x+6)}{2(x+6)(x-3)} = \frac{4x+1}{2(x-3)}$ and then use direct substitution.