Question on Limits

**h2osprey** · July 3rd 2008, 02:21 AM

Evaluate $\lim_{x \to \infty} \frac{(\frac{10}{11})^x}{(\frac{9}{10})^x + (\frac{11}{12})^x}$

Instinctively it's $\frac{1}{2}$ , just that I'm not too sure how to go about showing it.

**TwistedOne151** · July 3rd 2008, 03:04 AM

Note that 9/10, 10/11, and 11/12 are all less than one, so $\left(\frac{9}{10}\right)^x$ , $\left(\frac{10}{11}\right)^x$ , and $\left(\frac{11}{12}\right)^x$ all tend to zero as x goes to infinity. The largest of these numbers is 11/12, so it goes to zero most slowly. Thus, we multiply both the numerator and denominator by $\left(\frac{12}{11}\right)^x$ :
$\lim_{x\to\infty}\frac{(\frac{10}{11})^x}{(\frac{9 }{10})^x+(\frac{11}{12})^x}=\lim_{x\to\infty}\frac {(\frac{10}{11})^x(\frac{12}{11})^x}{((\frac{9}{10 })^x+(\frac{11}{12})^x)(\frac{12}{11})^x}$
$=\lim_{x\to\infty}\frac{(\frac{120}{121})^x}{(\fra c{54}{55})^x+1}$ .
Note that 120/121 and 54/55 are both less than one, as expected, and so as x goes to infinity, the numerator of the above goes to zero, and the denominator goes to one. Thus the limit is 0.

--Kevin C.

**wingless** · July 3rd 2008, 03:06 AM

$\lim_{x\to\infty} \frac{\left ( \frac{10}{11} \right )^x}{\left ( \frac{9}{10} \right )^x + \left ( \frac{11}{12} \right )^x}$

The general approach for these limits is factoring the largest term of the denominator. It is $\left ( \frac{11}{12} \right )^x$ here.

$\lim_{x\to\infty} \frac{\left ( \frac{11}{12} \right )^x \cdot \left ( \frac{120}{121} \right )^x}{\left ( \frac{11}{12} \right )^x \cdot \left [ 1 + \left ( \frac{108}{110} \right )^x \right ] }$

$\lim_{x\to\infty} \frac{\left ( \frac{120}{121} \right )^x}{1 + \left ( \frac{108}{110} \right )^x }$

We know that $\lim_{x\to\infty} r^x = 0$ for $|r|<1$ . So those fractions will go to zero.

$\lim_{x\to\infty} \frac{0}{1+0}=0$

**flyingsquirrel** · July 3rd 2008, 03:08 AM

Hello

Originally Posted by h2osprey

Evaluate $\lim_{x \to \infty} \frac{(\frac{10}{11})^x}{(\frac{9}{10})^x + (\frac{11}{12})^x}$

Instinctively it's $\frac{1}{2}$ , just that I'm not too sure how to go about showing it.

Another possible approach : $0<\frac{(\frac{10}{11})^x}{(\frac{9}{10})^x + (\frac{11}{12})^x}\leq\frac{(\frac{10}{11})^x}{(\f rac{11}{12})^x}=\left(\frac{120}{121}\right)^x\und erset{x\to\infty}{\to}0$ and we can conclude using the squeezing theorem.

Thread: Question on Limits

LinkBack

Thread Tools

Display

Question on Limits

Re: Limit

Similar Math Help Forum Discussions

Question about limits

Limits question

limits question

Limits question

Limits Question...

Search Tags