asymptotes of 7/(x-5) — step-by-step solution

Let's find the asymptotes of

\frac{7}{x - 5}

, for real

x

Vertical asymptotes: find the excluded values, where the denominator is zero. Then check whether the curve goes to infinity near these values

Let's solve in

ℝ

x - 5 = 0

ⓘ We add

5

to both sides

Solution obtained:

x = 5

x

approaches

5

from the left (smaller values), the function tends to

-∞

; from the right (larger values), it tends to

+∞

The function tends to infinity on at least one side: the line

x = 5

is therefore a vertical asymptote

At infinity, the numerator degree is smaller than the denominator degree: the line to test is

y = 0

For a horizontal asymptote, the function must tend to a finite value: check the limit of

\frac{7}{x - 5}

at both infinities

x

tends to +∞,

\frac{7}{x - 5}

tends to

0

; as

x

tends to -∞,

\frac{7}{x - 5}

tends to

0

The horizontal asymptote is

y = 0

; there is no slant asymptote

▶Asymptotes:

{x = 5, y = 0}

🖼/draw?f1=7%2F%28x+-+5%29&v=x&x2=5&f3=0&b=-5.0&e=15.0&min=-4.13&max=3.4299999999999997&dash=2,3