EVALUATING LIMITS AND FIND HORIZONTAL AND VERTICAL ASYMPTOTES

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.