IXL - Find limits at vertical asymptotes using graphs (Calculus practice)

The graph shows

(

)

The dashed line is a vertical asymptote.

Choose the correct statement.

lim

→1

(

)

–

∞

lim

→1

(

)

∞

ref_doc_title.

Learn with an example

question

The graph shows

(

)

The dashed line is a vertical asymptote.

Choose the correct statement.

lim

→3

(

)

–

∞

lim

→3

(

)

∞

key idea

The vertical line

is a vertical asymptote of

(

)

if at least one of the left-hand limit,

lim

→

(

)

or the right-hand limit,

lim

→

(

)

–

∞

or ∞.

The following rules describe what it means for a left-hand limit of a function

(

)

to be

–

∞

or ∞.

If the values of
g
(
x
)
become arbitrarily large positive numbers as
x
approaches
a
from the
left,
then
lim
x
→
a
-
g
(
x
)
=
∞
.
If the values of
g
(
x
)
become arbitrarily large negative numbers as
x
approaches
a
from the
left,
then
lim
x
→
a
-
g
(
x
)
=
–
∞
.

Corresponding rules apply for right-hand limits.

solution

The limit

lim

→3

(

)

has a raised minus sign after the 3. This means it is the left-hand limit of

(

)

at 3. To find the limit, look at the values of

(

)

approaches 3 from the left.

approaches 3 from the left, the values of

(

)

become arbitrarily large positive numbers. This means for any large positive number you can think of, there will be a point when

is close to 3 and the value of

(

)

will be larger than that number. Also, as

gets even closer to 3 from the left, all values of

(

)

will also be larger than that number.

So,

lim

→3

(

)

∞

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Work it out

Not feeling ready yet? These can help:

G.1Rational functions: asymptotes and excluded values

A.2Find one-sided limits using graphs

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