IXL - Determine the continuity of a piecewise function at a point (Calculus practice)

Is the function

(

)

continuous at

–

(

)

– 10 x – 16	if x ≤ – 1
– 3 x – 9	if x > – 1

yes

ref_doc_title.

Learn with an example

question

Is the function

(

)

continuous at

–

(

)

– 3	if x < – 2
8 x + 15	if x ≥ – 2

yes

key idea

A function

(

)

is continuous at

if all three of the following statements are true:

g
(
x
)
is defined at
x
=
a
.
The limit
lim
x
→
a
g
(
x
)
exists.
The limit equals the value of the function:
lim
x
→
a
g
(
x
)
=
g
(
a
)

solution

To determine whether

(

)

is continuous at

–

evaluate

(

–

)

and

lim

→

–

(

)

then check if they are equal.

To evaluate

(

–

)

look at the expression that defines

(

)

for

–

(

)

– 3	if x < – 2
8 x + 15	if x ≥ – 2

Plug

–

into this expression.

(

–

)

–

To evaluate

lim

→

–

(

)

look at the expression that defines

(

)

for values of

near, but not equal to,

–

Because there is more than one expression for

(

)

around

–

you must look at the one-sided limits individually.

You can start with the left-hand limit. To evaluate

lim

→

–

(

)

look at the expression that defines

(

)

for values of

near, but less than,

–

(

)

– 3	if x < – 2
8 x + 15	if x ≥ – 2

Use this expression to evaluate the left-hand limit.

lim

→

–

(

)

lim

→

–

Since

lim

→

–

(

)

≠

(

–

)

the function

(

)

is not continuous at

–

You don't have to evaluate the right-hand limit. This is because for

lim

→

–

(

)

to equal

(

–

)

both one-sided limits must equal

(

–

)

Looking for more? Try this:

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Not feeling ready yet? These can help:

B.6Find limits involving factorization and rationalization

E.2Determine continuity using graphs

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