asymptotes of f(x)=(3x-15)/(x^2-25)

Let's find the asymptotes of

\frac{3 x - 15}{x^{2} - 25}

, 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^{2} - 25 = 0

The two numbers whose square is

25

are

-5

and

5

Solutions found:

x = {-5, 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

x

approaches

5

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

\frac{3}{10}

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

\frac{3}{10}

The limits at

5

are finite: this is a hole, not 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{3 x - 15}{x^{2} - 25}

at both infinities

x

tends to +∞,

\frac{3 x - 15}{x^{2} - 25}

tends to

0

; as

x

tends to -∞,

\frac{3 x - 15}{x^{2} - 25}

tends to

0

The horizontal asymptote is

y = 0

; there is no slant asymptote

▶Asymptotes:

{x = -5, y = 0}

🖼/draw?f1=%283*x+-+15%29%2F%28x%5E2+-+25%29&v=x&x2=-5&f3=0&b=-15.0&e=5.0&min=-1.77&max=1.47&dash=2,3