How do you find vertical, horizontal and oblique asymptotes for #f(x) = (5x-15 )/ (2x+14)#?

Question

Lily Anderson · Accepted Answer

Vertical asymptote = x = -7, Horizontal asymptote = y = 5/2, no oblique asymptotes.

An ASYMPTOTE is a line that approches a curve, but NEVER meets it.

To find the vertical asymptote , put the denominator = 0 (because 0 cannot divide any number) and solve.

Given below is the step-by-step walk through

The curve can never touch x = -7, thus making it the vertical asymptote.

To find the horizontal asymptote , compare the degree of the expressions in the numerator and the denominator.
Here, the degree of the numerator = 1 and the degree of the denominator = 1.
Since the degrees are equal, the horizontal asymptote

#y = ("numerator's leading coefficient)"/"(denominator's leading coefficient")#

# y = 5/2#

The oblique asymptote is a line of the form y = mx + c.
In other words,
degree of numerator = degree of denominator + 1
Here, the degree of the numerator = degree of the denominator.
Therefore, the given function has no oblique asymptotes.

The graph for the function is given below:
graph{(5x-15)/(2x+14) [-35.9, 37.16, -15.5, 21.04]}![enter image source here]

Madelyn Anderson · Answer

undefined To find the vertical asymptotes of a rational function, set the denominator equal to zero and solve for $ x $. These values of $ x $ will give the vertical asymptotes.

For horizontal asymptotes:
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $ y = 0 $.
- If the degrees are equal, divide the leading coefficient of the numerator by the leading coefficient of the denominator to find the horizontal asymptote.

To find the oblique (slant) asymptote:
- If the degree of the numerator is exactly one more than the degree of the denominator, divide the numerator by the denominator using long division or polynomial division. The quotient obtained will be the equation of the oblique asymptote.

For the function $ f(x) = \frac{5x - 15}{2x + 14} $:

Vertical asymptote: Set the denominator equal to zero and solve for $ x $:
$$ 2x + 14 = 0 $$
$$ 2x = -14 $$
$$ x = -7 $$
So, there is a vertical asymptote at $ x = -7 $.

Horizontal asymptote: The degrees of the numerator and denominator are equal, so we divide the leading coefficient of the numerator by the leading coefficient of the denominator:
$$ \frac{5}{2} $$
So, the horizontal asymptote is $ y = \frac{5}{2} $.

Oblique asymptote: Since the degree of the numerator is exactly one more than the degree of the denominator, we perform polynomial division:
$$ \frac{5x - 15}{2x + 14} = \frac{5}{2} - \frac{55}{4(x + 7)} $$
Thus, the equation of the oblique asymptote is $ y = \frac{5}{2} $.