Determine if the limit exists, and if it does, find its value?

Question

lim 4x+3/ ( x^2-2x+5) as x approaches infinity..
A) lim=3
B) lim=5
C) lim=-2
D) lim does not exist
E) lim= 3/5
F) lim= 0

Answer 1

Hi,

The correct answer is F. When you have a rational function and you are looking for the end behavior as x approaches infinity, here are a couple real basic rules:

The degree of a polynomial is the largest power it would have if all factors were multiplied together in the numerator or denominator. If you compare the degree of the numerator with the degree of the denominator and the degree of the numerator is smaller like this problem, then the x values always approach 0 as you go to infinity. Realize that a fraction like
4(100) + 3
============== would be very small with big denominator
100^2 -2(100) + 5

On the other hand, if the degree on the top and the bottom is the same, then as x approaches infinity, the graph approaches the number you get by dividing the leading coefficients by each other. So if you had:

8x^2 - 10x + 3
============ the leading coefficients are 8 and -2
5 - 2x^2 because those are coefficients in terms
with the largest exponents. That means
that as x goes to infinity, the graph approaches 8/-2 = -4.

If the exponent was larger on top by one, you have a slant asymptote and you need to divide the top expression by the bottom expression and discard any remainder to find the equation of the slant asymptote that the graph will follow as it approaches infinity.

I hope this helps.

Answer 2

lim (4x + 3) / (x^2 - 2x + 5)
x -> infinity

Divide top and bottom by the highest power of x.

lim ( [4/x + 3/x^2] / [1 - 2/x + 5/x^2] )
x -> infinity

Evaluate term by term.

[0 + 0] / [1 - 0 + 0] = 0/1 = 0