Limit goes to $\infty$, but wolfram alpha says its limit is $0$

Question

I have $(5/3)^n$, if I input this in wolframalpha.com I can see it goes to infinity as long as n increases, but if I go down in the wolframalpha webpage results it shows .... the limit is zero as long as n increases, am I wrong?

Answer 1

As you've correctly said, the value of the limit approaches $\infty$ as $n \to \infty$.

I typed the limit into Wolfram|Alpha using these inputs:

limit n to infinity (5/3)^n

lim_{n to infinity} (5/3)^n

And they both gave me the correct answers.

What I suspect you've done is this:

limit n to infinity 5/3^n

lim_{n to infinity} 5/3^n

Which is $\lim_{n \to \infty} \frac{5}{3^n}=0$ instead of the correct limit $\lim_{n \to \infty} \left(\frac{5}{3}\right)^n=\lim_{n \to \infty} \left(\frac{5^n}{3^n}\right)=\infty$

Edit: After you provided the screenshot:

Look carefully at the input given by Wolfram Alpha on your screenshot.

$$\lim_{\color{red}{n \to -\infty}} \left(\frac{5}{3}\right)^n=0$$

The limit tends to $-\infty$ instead of $+\infty$.

To avoid this problem in the future: Type the limit explicitly rather than just typing $(\frac{5}{3})^n$ on Wolfram|Alpha as I have done on my top two examples.

Answer 2

This sequence definitely approaches infinity as $n$ goes to infinity.

Answer 3

I suspect you are missing a pair of parentheses: If you type

Limit[5/3^n, n -> Infinity]

You will get 0, because the order of operations makes this equivalent to $$\lim_{n \to \infty} \frac{5}{3^n} = 0.$$ If you type

Limit[(5/3)^n, n -> Infinity]

you will get your intended result, which is $$\lim_{n \to \infty} \left(\frac{5}{3}\right)^n = \infty.$$

Answer 4

You said down the page was a limit of 0. That's because they took $n -> -\infty$. Edit OP shows the same thing circled on screen print.