Thread: Limit of an exponential function?

1. Limit of an exponential function?

Can you help me, I don't know how to find the limit for this function, in this picture: (It's for a practice quiz and only for a participation grade, but similar questions are going to show up later on tests and I'm not sure how to solve them) Thanks!

2. Re: Limit of an exponential function?

Why don't you know how to find the limit for that function? Use your limit laws. Show some effort.

3. Re: Limit of an exponential function?

Originally Posted by SlipEternal
Why don't you know how to find the limit for that function? Use your limit laws. Show some effort.
I tried to, but I got 0 for the limit to infinity, and 0 for the limit to negative infinity! And I don't think that's right. And for the limit to 0 for the last one I got 1200 exactly, it just seems like I must be doing something wrong, because I didn't get any of the options! :/

4. Re: Limit of an exponential function?

Originally Posted by canyouhelp
I tried to, but I got 0 for the limit to infinity, and 0 for the limit to negative infinity! And I don't think that's right. And for the limit to 0 for the last one I got 1200 exactly, it just seems like I must be doing something wrong, because I didn't get any of the options! :/
given $0<a<1$ what is

$\displaystyle{\lim_{x \to \infty}} a^x$ ?

$\displaystyle{\lim_{x \to -\infty}} a^x$ ?

$\displaystyle{\lim_{x \to 0}} ~a^x$

if you can answer these then the limit of f(x) in these cases becomes obvious.

5. Re: Limit of an exponential function?

Originally Posted by romsek
given $0<a<1$ what is

$\displaystyle{\lim_{x \to \infty}} a^x$ ?

$\displaystyle{\lim_{x \to -\infty}} a^x$ ?

$\displaystyle{\lim_{x \to 0}} ~a^x$

if you can answer these then the limit of f(x) in these cases becomes obvious.
If I set a to .5 then I get x's limit to infinity is 100, so III is right. X's limit to -infinity is 100 too, so IV is right. The limit to 0 is about 96, but V is about 93 isn't it? So it still doesn't fit any of my options. Am I calculating something wrong?

6. Re: Limit of an exponential function?

$0.5^2 = 0.25<0.5$. So, the function is decreasing. It seems unlikely that $\lim_{x \to \infty} (0.5)^x = 100$. How did you arrive at that answer? Note that if $0<a<1$ then there exists a number $1<b$ such that $a = \dfrac{1}{b}$. Then $a^x = \left(\dfrac{1}{b}\right)^x = \dfrac{1}{b^x}$. You should know that for any $1<b$, $\lim_{x \to \infty} b^x = \infty$. So, as $x$ approaches infinity, the denominator approaches infinity, and the whole quotient approaches 0.

Next, $\lim_{x \to -\infty} a^x = \lim_{x \to \infty} a^{-x} = \lim_{x \to \infty} b^x = \infty$.

Finally, $a^x$ is a continuous function, so $\lim_{x \to 0} a^x = a^0 = 1$, regardless of the value of $a$. You definitely need practice evaluating exponents.

\begin{align*}\lim_{x \to \infty} a^x & = \lim_{x\to \infty} \dfrac{1}{b^x} \\ & = \dfrac{1}{\lim_{x \to \infty} b^x}\end{align*}

In other words, I get the correct answer to the original problem is B. Only I and IV.

7. Re: Limit of an exponential function?

Originally Posted by canyouhelp
If I set a to .5 then I get x's limit to infinity is 100, so III is right. X's limit to -infinity is 100 too, so IV is right. The limit to 0 is about 96, but V is about 93 isn't it? So it still doesn't fit any of my options. Am I calculating something wrong?
Using examples is a great way to develop intuition, but it may sometimes lead you astray, and it certainly does not constitute proof.

romsek said above that you should apply the limit laws. That was great advice. You need to memorize them and learn to apply them.

$\displaystyle \lim_{x \rightarrow a}\{f(x) \pm g(x)\} = \{\lim_{x \rightarrow a}f(x)\} \pm \{\lim_{x \rightarrow a}g(x)\}.$ Addition/subtraction Law.

$\displaystyle \lim_{x \rightarrow a}\{f(x) * g(x)\} = \{\lim_{x \rightarrow a}f(x)\} * \{\lim_{x \rightarrow a}g(x)\}.$ Multiplication Law.

$\displaystyle \lim_{x \rightarrow a}g(x)\ is\ a\ real\ number \implies \lim_{x \rightarrow a}\left(\dfrac{f(x)}{g(x)}\right) = \dfrac{\displaystyle \lim_{x \rightarrow a}f(x)}{\displaystyle \lim_{x \rightarrow a}g(x)}.$ Division Law

So the question is trivial except for $\displaystyle \lim_{x \rightarrow c}a^x,\ given\ 0 < a < 1.$