1. ## limits / equations?

for what value of a is the following equation true?

lim ((x+a)/(x-a))^x = e
x->inf

2. ## Re: limits / equations?

Take the natural log of both sides, so that you wind up with:

$\lim_{x\to\infty}\frac{\ln\left(\frac{x+a}{x-a} \right)}{\frac{1}{x}}=1$

Now use L'Hôpital's rule to find $a$.

3. ## Re: limits / equations?

after applying L'Hôpital's rule i got the limit as x approaches infinity ((x-a)(-2a))/(-x^-2) = 1
and now what do i do from there?

4. ## Re: limits / equations?

You have differentiated the numerator incorrectly. You want:

$\frac{d}{dx}\left(\ln\left(\frac{x+a}{x-a} \right) \right)=\frac{1}{\frac{x+a}{x-a}}\cdot\frac{-2a}{(x-a)^2}=\frac{2a}{a^2-x^2}$

Now, putting this together with the derivative of the denominator, we have:

$\lim_{x\to\infty}\frac{2ax^2}{x^2-a^2}=1$

which we can write as:

$2a\lim_{x\to\infty}\frac{1}{1-\frac{a^2}{x^2}}=1$

From here, it is a piece of cake.

5. ## Re: limits / equations?

im not seeing how this is a piece of cake?

6. ## Re: limits / equations?

Originally Posted by pnfuller
im not seeing how this is a piece of cake?
LEARN this: $\lim _{x \to \infty } \left( {1 + \frac{a}{{x + b}}} \right)^{cx} = e^{ac}$

Yours can be rewritten an $\left( {1 + \frac{{2a}}{{x - a}}} \right)^x$.

7. ## Re: limits / equations?

What is $\lim_{x\to\infty}\frac{a^2}{x^2}$ ?

8. ## Re: limits / equations?

the limit as a^2/x^2 approaches infinity equals 0? is that what you mean?

9. ## Re: limits / equations?

Yes, so what does that tell you about the limit at the end of post #4?

10. ## Re: limits / equations?

that the limit as x approaches infinity is 1/1 and it equals 1 but what does a equal?

11. ## Re: limits / equations?

Reread that equation carefully. You were told \displaystyle \begin{align*} 2a \cdot \lim_{x \to \infty}\frac{1}{1 - \frac{a^2}{x^2}} = 1 \end{align*}

12. ## Re: limits / equations?

Okay, since the limit is 1, you simply have:

$2a=1$

Now solve for $a$.

so a = 1/2

14. ## Re: limits / equations?

Originally Posted by pnfuller
that the limit as x approaches infinity is 1/1 and it equals 1 but what does a equal?
This whole thread is totally frustrating for me.
The answer is $a=0.5$ . WHY?