Question

For the following exercises, make tables to show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote.

f(x) = 2x/(x + 4)

Answer 1

To show the behavior of the function near the vertical asymptote and reflecting the horizontal asymptote:

f(x) = 2x/(x + 4)

 

A vertical asymptote occurs at the zeros of the factors of the denominator.

To find the vertical asymptote:

It is seen that the denominator of the given fraction x + 4 will be zero at x = -4.

So there is a vertical asymptote at x = -4.

 

Table showing the values of f(x) when the input values are smaller than -4 and become negative:

x	-4.5	-5	-5.5	-6	-7
f(x) = 2x/(x + 4)	18	10	7.33	6	4.67

 

It can be seen from the table that as the x → 4-, f(x) → ∞.

 

Table showing the values of f(x) when the input values are greater than -4 and become larger:

x	-3.5	-3	-2.5	-2	-1
f(x) = 2x/(x + 4)	-14	-6	-3.33	-2	-0.67

 

It can be seen from the table that as x → 4+, f(x) → ∞.

To find the horizontal asymptote:

Relate the given function f(x) = 2x/(x + 4) with f(x) = p(x)/q(x), q(x) ≠ 0.

 

So it is seen that degree of p = 1 and degree of q = 1.

Since p = q, there is a horizontal asymptote found at 2x/(x + 4).

Horizontal asymptote is the ratio of the leading coefficients of numerator and denominator.

So, horizontal asymptote is at y = 2/1 or y = 2.

So, horizontal asymptote is at y = 2/1 or y = 2.