# EVALUATING LIMITS AND FIND HORIZONTAL AND VERTICAL ASYMPTOTES

A function f(x) will have the horizontal asymptote y = L if either

lim x→∞  f(x) = L or lim x→−∞ f(x) = L

To find horizontal asymptotes, we simply evaluate the limit of the function as it approaches infinity, and again as it approaches negative infinity.

Evaluate each limit and then identify any horizontal asymptotes.

Problem 1 :

Solution:

Horizontal asymptote:

Degree of numerator = 3

Degree of denominator = 3

degree of numerator = degree of denominator

y = leading coefficient of N(x) / coefficient of D(x)

y = 1/2

So, equation of the horizontal asymptote is y = 1/2.

Problem 2 :

Solution:

Horizontal asymptote:

Degree of numerator = 2

Degree of denominator = 3

degree of numerator < degree of denominator

So, equation of the horizontal asymptote is y = 0 which is the x-axis.

Problem 3 :

Solution:

Horizontal asymptote:

Degree of numerator = 5

Degree of denominator = 3

degree of numerator > degree of denominator

So, there is no horizontal asymptote. Slant  asymptote  will be there.

Problem 4 :

Solution:

Horizontal asymptote:

Degree of numerator = 1

Degree of denominator = 2

degree of numerator < degree of denominator

So, equation of the horizontal asymptote is y = 0 which is the x-axis.

Problem 5 :

Solution:

Horizontal asymptote:

Degree of numerator = 2

Degree of denominator = 1

degree of numerator > degree of denominator

So, there is no horizontal asymptote.

Evaluate each limit and then identify any vertical asymptotes.

Problem 6 :

Solution:

Vertical asymptote:

Equate the denominator to zero and solve for x.

x - 2 = 0

x = 2

So, the equation of the vertical asymptote is x = 2.

Problem 7 :

Solution:

Vertical asymptote:

Equate the denominator to zero and solve for x.

(x - 1)3 = 0

x = 1

So, the equation of the vertical asymptote is x = 1.

Problem 8 :

Solution:

Vertical asymptote:

Equate the denominator to zero and solve for x.

x2 - 9 = 0

x2 = 9

x = 3

So, the equation of the vertical asymptote is x = 3.

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