Question

Evaluate the limit as x -> a of [f(x)-f(a)]/[sqroot(x)-sqroot(a)]. The limit is a derivative of a function, and the derivative must be used to evaluate the function.

Answer 1

The first thing I did was to try and eliminate the squareroot(x)-sqroot(a) on the bottom by multiplying by the conjugate. But none of that worked, as the denominators all had the same problem..

Answer 2

use l'hopital rule

Answer 3

Multiplying by conjugate should have worked.

Answer 4

$$\begin{align*}
\lim \limits_{x\to a}\frac{f(x)-f(a)}{\sqrt x-\sqrt a}&=\lim \limits_{x \to a}\frac{f(x)-f(a)}{x-a}\left(\sqrt x+\sqrt a\right)\\
&=f'(x)\left[\lim \limits_{x \to a}\sqrt x+\sqrt a\right]\\&=2\sqrt a f'(x)\end{align*}$$

Answer 5

Ohhhhhh... that middle part equals f'(x) because that's the general rule for it. So then it would 2root(a) times f'(a). But wouldn't that still give my indeterminate form?

Answer 6

or would my final answer just be 2root(a)?

Answer 7

answer should depend on the function. [ btw your right final answer should be 2sqrt(a)f'(a) not 2 sqrt(a)f'(x)]

Answer 8

right but wouldn't f'(a) just equal 0/0 again?

Answer 9

no it depends on f(x). Is f(x) given?.

If not
say f(x)=qx^2+q
f'(x)=2qx, f'(a)=2aq

the answer is 2sqrt(a)(2aq). Just like that for another function f(x) there will be another solution.

Answer 10

Got it! Thanks!

Answer 11

you're welcome:)