Popular Problems Calculus Evaluate limit as x approaches 0 of (tan(x))/x
lim
x
→
0
tan
(
x
)
x
Evaluate the limit of the numerator and the limit of the denominator.
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Take the limit of the numerator and the limit of the denominator.
lim
x
→
0
tan
(
x
)
lim
x
→
0
x
Evaluate the limit of the numerator.
Tap for fewer steps...
Move the limit inside the trig function because tangent is continuous.
tan
(
lim
x
→
0
x
)
lim
x
→
0
x
Evaluate the limit of
x
by plugging in
0
for
x
.
tan
(
0
)
lim
x
→
0
x
The exact value of
tan
(
0
)
is
0
.
0
lim
x
→
0
x
Evaluate the limit of
x
by plugging in
0
for
x
.
0
0
The expression contains a division by
0
The expression is undefined.
Undefined
Since
0
0
is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
lim
x
→
0
tan
(
x
)
x
=
lim
x
→
0
d
d
x
[
tan
(
x
)
]
d
d
x
[
x
]
Find the derivative of the numerator and denominator.
Tap for fewer steps...
Differentiate the numerator and denominator.
lim
x
→
0
d
d
x
[
tan
(
x
)
]
d
d
x
[
x
]
The derivative of
tan
(
x
)
with respect to
x
is
sec
2
(
x
)
.
lim
x
→
0
sec
2
(
x
)
d
d
x
[
x
]
Differentiate using the Power Rule which states that
d
d
x
[
x
n
]
is
n
x
n
−
1
where
n
=
1
.
lim
x
→
0
sec
2
(
x
)
1
Take the limit of each term.
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Split the limit using the Limits Quotient Rule on the limit as
x
approaches
0
.
(
lim
x
→
0
sec
2
(
x
)
)
(
lim
x
→
0
1
)
Move the exponent
2
from
sec
2
(
x
)
outside the limit using the Limits Power Rule.
(
lim
x
→
0
sec
(
x
)
)
2
lim
x
→
0
1
Move the limit inside the trig function because secant is continuous.
Answers & Comments
Answer:
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Calculus Examples
Popular Problems Calculus Evaluate limit as x approaches 0 of (tan(x))/x
lim
x
→
0
tan
(
x
)
x
Evaluate the limit of the numerator and the limit of the denominator.
Tap for fewer steps...
Take the limit of the numerator and the limit of the denominator.
lim
x
→
0
tan
(
x
)
lim
x
→
0
x
Evaluate the limit of the numerator.
Tap for fewer steps...
Move the limit inside the trig function because tangent is continuous.
tan
(
lim
x
→
0
x
)
lim
x
→
0
x
Evaluate the limit of
x
by plugging in
0
for
x
.
tan
(
0
)
lim
x
→
0
x
The exact value of
tan
(
0
)
is
0
.
0
lim
x
→
0
x
Evaluate the limit of
x
by plugging in
0
for
x
.
0
0
The expression contains a division by
0
The expression is undefined.
Undefined
Since
0
0
is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
lim
x
→
0
tan
(
x
)
x
=
lim
x
→
0
d
d
x
[
tan
(
x
)
]
d
d
x
[
x
]
Find the derivative of the numerator and denominator.
Tap for fewer steps...
Differentiate the numerator and denominator.
lim
x
→
0
d
d
x
[
tan
(
x
)
]
d
d
x
[
x
]
The derivative of
tan
(
x
)
with respect to
x
is
sec
2
(
x
)
.
lim
x
→
0
sec
2
(
x
)
d
d
x
[
x
]
Differentiate using the Power Rule which states that
d
d
x
[
x
n
]
is
n
x
n
−
1
where
n
=
1
.
lim
x
→
0
sec
2
(
x
)
1
Take the limit of each term.
Tap for fewer steps...
Split the limit using the Limits Quotient Rule on the limit as
x
approaches
0
.
(
lim
x
→
0
sec
2
(
x
)
)
(
lim
x
→
0
1
)
Move the exponent
2
from
sec
2
(
x
)
outside the limit using the Limits Power Rule.
(
lim
x
→
0
sec
(
x
)
)
2
lim
x
→
0
1
Move the limit inside the trig function because secant is continuous.
sec
2
(
lim
x
→
0
x
)
lim
x
→
0
1
Evaluate the limits by plugging in
0
for all occurrences of
x
.
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Evaluate the limit of
x
by plugging in
0
for
x
.
sec
2
(
0
)
lim
x
→
0
1
Evaluate the limit of
1
which is constant as
x
approaches
0
.
sec
2
(
0
)
1
Simplify the answer.
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Divide
sec
2
(
0
)
by
1
.
sec
2
(
0
)
The exact value of
sec
(
0
)
is
1
.
1
2
One to any power is one.
1