asymptotes of 1/(x-1)^2 — step-by-step solution

Let's find the asymptotes of

\frac{1}{{(x - 1)}^{2}}

, 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 - 1)}^{2} = 0

{(x - 1)}^{2} = 0

is equivalent to:

x - 1 = 0

ⓘ We add

1

to both sides

Solution found:

x = 1

x

approaches

1

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 = 1

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{1}{{(x - 1)}^{2}}

at both infinities

x

tends to +∞,

\frac{1}{{(x - 1)}^{2}}

tends to

0

; as

x

tends to -∞,

\frac{1}{{(x - 1)}^{2}}

tends to

0

The horizontal asymptote is

y = 0

; there is no slant asymptote

▶Asymptotes:

{x = 1, y = 0}

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