Limits

**Mathsnewbie** · May 5th 2010, 10:46 AM

Can someone help me with this problem:

lim (n^2 + n)^(1/n) = 1;

In (b) you may assume that n^1/n = 1.

I know how to solve these when it's a number to the power of n but not when it's n to a power.

**JG89** · May 5th 2010, 11:22 AM

$(n^2 + n)^{1/n} = [n(n+1)]^{1/n} = n^{1/n} (n+1)^{1/n}$ .

Can you show that $(n+1)^{1/n}$ goes to 1 as n goes to infinity?

**tonio** · May 5th 2010, 11:24 AM

Originally Posted by Mathsnewbie

Can someone help me with this problem:

lim (n^2 + n)^(1/n) = 1;

In (b) you may assume that n^1/n = 1.

What does this mean??

I know how to solve these when it's a number to the power of n but not when it's n to a power.

If you already know $\sqrt[n]{n}\xrightarrow [n\to\infty]{}1$ , then it's easy: $\sqrt[n]{n^2+n}=\sqrt[n]{n}\sqrt[n]{n+1}\xrightarrow [n\to\infty]{}1\cdot 1=1$ .

Tonio

**Mathsnewbie** · May 5th 2010, 11:25 AM

Originally Posted by JG89

$(n^2 + n)^{1/n} = [n(n+1)]^{1/n} = n^{1/n} (n+1)^{1/n}$ .

Can you show that $(n+1)^{1/n}$ goes to 1 as n goes to infinity?

I can't, I know how to do some by sandwich principle but I couldn't get either of these to work.

**tonio** · May 5th 2010, 11:28 AM

Originally Posted by Mathsnewbie

I can't, I know how to do some by sandwich principle but I couldn't get either of these to work.

$\sqrt[n]{a}\xrightarrow [n\to\infty]{}1\,,\,\,\forall a>0$ , so just check that $1\leq\sqrt[n]{n+1}\leq \sqrt[n]{2n}=\sqrt[n]{2}\,\sqrt[n]{n}$ ...

Tonio

**Mathsnewbie** · May 5th 2010, 11:34 AM

Thank you, I managed to complete

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