MHB Show $\displaystyle \lim_{x \to \infty} \frac{50x^{10}+100}{x^{11}+x^6+1}=0$

[Guest2]

How do you show that $\displaystyle \lim_{x \to \infty} \frac{50x^{10}+100}{x^{11}+x^6+1}=0$

What I tried:

$\displaystyle \lim_{x \to \infty} \frac{50x^{10}+100}{x^{11}+x^6+1} =\lim_{x \to \infty} \frac{50+100/x^{11}}{1+1/x^{5}+1/x^{11}} = \frac{50+0}{1+0+0} = 50.$

But this is wrong. (Angry)

 

[evinda]

If you have a fraction of polynomials and you want to find the limit as $x \to +\infty$ you look only at the terms of largest degree.

So your limit is equal to $\lim_{x \to +\infty} \frac{50 x^{10}}{x^{11}}=\lim_{x \to +\infty} \frac{50}{x}=0$

 

[Guest2]

If you have a fraction of polynomials and you want to find the limit as $x \to +\infty$ you look only at the terms of largest degree.

So your limit is equal to $\lim_{x \to +\infty} \frac{50 x^{10}}{x^{11}}=\lim_{x \to +\infty} \frac{50}{x}=0$

Thanks. When is it that we divide the highest power then?

 

[evinda]

Thanks. When is it that we divide the highest power then?

You could also divide by the highest power in this case, if you would want to. The result will be the same...

 

[Guest2]

You could also divide by the highest power in this case, if you would want to. The result will be the same...

I get it now - my algebra was wrong in the original post.

$\displaystyle \lim_{x \to \infty} \frac{50x^{10}+100}{x^{11}+x^6+1} =\lim_{x \to \infty} \frac{50/x+100/x^{11}}{1+1/x^{5}+1/x^{11}} = \frac{0+0}{1+0+0} = 0$

Thanks, again.