# Limits probably involving e^x-1/x special limit

## Homework Statement

lim of (a^x-a^-x-2)/(x^2) as x->0

Find the value of the limit^.

I know how to get the answer using L'Hospital.
However, I do not want to use Hospital's rule.
I think I see the pattern with the special limit:
lim of (e^x-1)/x as x->0

But I just don't see how I can apply that. I tried to take ln() out of everything, but it leads me to nowhere.

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Dick
Homework Helper

## Homework Statement

lim of (a^x-a^-x-2)/(x^2) as x->0

Find the value of the limit^.

I know how to get the answer using L'Hospital.
However, I do not want to use Hospital's rule.
I think I see the pattern with the special limit:
lim of (e^x-1)/x as x->0

But I just don't see how I can apply that. I tried to take ln() out of everything, but it leads me to nowhere.
a^x=e^(ln(a)x) is the connection between powers of e and a. You could also substitute the series expansions of the function. But I don't think knowing the special limit you've got there will help.

Then, how else could we solve this problem?
My school hasn't taught me Hospital's rule. So, there must be another way.

As far as I can see, L'Hopital's rule doesn't even apply since the numerator does not equal 0 if you plug in 0. a^0 - a^(-0) - 2 = 1 - 1 - 2 = -2.

You're simply dividing by a small positive number, so the limit is negative infinity.

vela
Staff Emeritus
Homework Helper
I think there's a typo in the original post. The problem should be
$$\lim_{x \to 0} \frac{a^x + a^{-x} - 2}{x^2}.$$ You can rewrite this as
$$\lim_{x \to 0} \frac{a^{-x}[(a^x)^2 - 2a^x +1]}{x^2}.$$ You can do a little algebra, use what Dick said about expressing this in terms of $e$, and change variables appropriately to relate it back to the special limit.

Dick
$$\lim_{x \to 0} \frac{a^x + a^{-x} - 2}{x^2}.$$ You can rewrite this as
$$\lim_{x \to 0} \frac{a^{-x}[(a^x)^2 - 2a^x +1]}{x^2}.$$ You can do a little algebra, use what Dick said about expressing this in terms of $e$, and change variables appropriately to relate it back to the special limit.