Evaluate Limit: 3x^3 + x + 26 / 20x^2 - 5x^3

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In summary, the conversation is about evaluating a limit and finding the correct answer to be -3/5. The method used is dividing the numerator and denominator by the highest exponent of x, but L'Hopital's Rule can also be used. The general method for finding the limit of a quotient of two polynomials is also mentioned, with three cases depending on the degrees of the polynomials.
  • #1
rcmango
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Homework Statement



Evaluate the Limit.

lim x -> infinity (3x^3 + x + 26) / (20x^2 - 5x^3)

Homework Equations





The Attempt at a Solution



I found the answer to be -3/5. is this correct?
I just divided the numerator and the denominator by the greatest exponent.
I see how it was done, but what are the rules to such a problem, and maybe someone could explain what is really going on here. Thankyou.
 
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  • #2
rcmango said:

Homework Statement



Evaluate the Limit.

lim x -> infinity (3x^3 + x + 26) / (20x^2 - 5x^3)

Homework Equations


The Attempt at a Solution



I found the answer to be -3/5. is this correct?

yes.

I just divided the numerator and the denominator by the greatest exponent.
I see how it was done, but what are the rules to such a problem, and maybe someone could explain what is really going on here. Thankyou.

to just look and see is the best way. If you wanted, I guess you could have used L'Hopital's Rule:

f=3x^3 + x + 26
g=20x^2 - 5x^3

lim f/g -> inf/inf
lim f'/g' -> inf/inf
lim f''/g'' -> inf/inf
lim f'''/g''' -> -3/5
 
  • #3
okay i see, but in the original method, do i divide by the largest exponent in the numerator or the denominator, i know I divide the top and bottom by this exponent.
 
  • #4
There is a general method for the limit of the quotient of 2 polynomials. Write out the quotient of a general polynomial of degree m, co efficients are a_m, a_{m-1}..., and then divide by another polynomial degree n, co efficients are b_n, b_{n-1}. There are 3 cases, 1) m > n, m=n and m< n. For every case, divide through by the highest exponent of x. What do you get?
 

What is the limit of the given function as x approaches infinity?

The limit of the given function as x approaches infinity is 0. This can be determined by dividing the highest degree term in the numerator (3x^3) by the highest degree term in the denominator (5x^3) and simplifying.

How can the limit of the given function be evaluated?

The limit of the given function can be evaluated by factoring out the highest degree term in the denominator and then simplifying the resulting fraction.

What is the limit of the given function as x approaches 0?

The limit of the given function as x approaches 0 is undefined. This is because the denominator becomes 0, which is not a valid number in mathematics.

What is the significance of finding the limit of a function?

Finding the limit of a function can help determine the behavior of the function at certain points and can provide insight into the overall behavior of the function. It can also be used to evaluate the continuity of a function.

How does the graph of the given function relate to its limit?

The limit of a function represents the value that the function approaches as the input approaches a certain point. This can be visualized on a graph by looking at the behavior of the function as the input gets closer and closer to the point in question.

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