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

In summary, when finding the limit of a fraction of polynomials as x approaches infinity, one only needs to consider the terms with the highest degree. In this case, the limit is equal to 0.
  • #1
Guest2
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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)
 
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  • #2
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 $\lim_{x \to +\infty} \frac{50 x^{10}}{x^{11}}=\lim_{x \to +\infty} \frac{50}{x}=0$
 
  • #3
evinda said:
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?
 
  • #4
Guest said:
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...
 
  • #5
evinda said:
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.
 

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

What is a limit?

A limit in mathematics is the value that a function approaches as its input (such as x) approaches a certain value (such as infinity).

How do you calculate a limit?

To calculate a limit, you can use techniques such as substitution, factoring, or the squeeze theorem. In this case, you can divide the numerator and denominator by the highest power of x to simplify the expression and then substitute infinity for x.

What does it mean when the limit of a function is 0?

If the limit of a function is 0, it means that the function approaches 0 as its input approaches a certain value. In other words, the function gets closer and closer to 0, but may never actually reach it.

Why is it important to find the limit of a function?

Finding the limit of a function can help us understand the behavior of the function as its input approaches a certain value. It can also help us determine if a function is continuous or if it has any asymptotes.

What does it mean when the limit of a function is undefined?

If the limit of a function is undefined, it means that the function does not approach a specific value as its input approaches a certain value. This can occur when there is a vertical asymptote, a discontinuity, or the function oscillates between two values as it approaches the limit.

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