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Trigonometric Limits: Combining techniques
Related Topics
Wize University Calculus 1 Textbook > Limits
The Fundamental Trig Limit
2 Activities
Evaluate
lim
x
→
0
x
ln
(
1
x
+
2
x
)
e
1
+
tan
x
+
ln
x
(
1
+
arctan
x
)
sin
x
+
x
cos
x
\lim\limits_{x\rightarrow0}\frac{x\ln(\frac{1}{x+2^x})e^{{1+\tan\ x+\ln\ x}}}{(1+\arctan\ x)\sin\ x+x\cos\ x}
x
→
0
lim
(
1
+
a
r
c
t
a
n
x
)
s
i
n
x
+
x
c
o
s
x
x
l
n
(
x
+
2
x
1
)
e
1
+
t
a
n
x
+
l
n
x
Answer
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Check Submission
More The Fundamental Trig Limit Questions:
Trig Limits
Find the value of
lim
x
→
0
sin
2
x
sin
3
x
\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}
x
→
0
lim
sin
3
x
sin
2
x
.
Fundamental trigonometric limits
Evaluate
lim
x
→
0
tan
x
−
sin
x
x
cos
x
\lim\limits_{x\to0}\frac{\tan x-\sin x}{x\cos x}
x
→
0
lim
x
c
o
s
x
t
a
n
x
−
s
i
n
x
.
Fundamental trigonometric limits
Evaluate
lim
x
→
0
tan
x
−
sin
x
x
cos
x
\lim\limits_{x\to0}\frac{\tan x-\sin x}{x\cos x}
x
→
0
lim
x
c
o
s
x
t
a
n
x
−
s
i
n
x
.
Fundamental trigonometric limits
Determine
lim
x
→
0
x
2
tan
2
2
x
\lim\limits_{x\rightarrow0}\frac{x^2}{\tan^2\ 2x}
x
→
0
lim
t
a
n
2
2
x
x
2
.
Trig Limits
Find the value of
lim
x
→
0
sin
2
x
sin
3
x
\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}
x
→
0
lim
sin
3
x
sin
2
x
.
Trigonometric Limits
Practice: Trig Limits
Evaluate
lim
x
→
0
2
sin
5
x
x
2
\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}
x
→
0
lim
x
2
2
sin
5
x
.
Trig Limits
Find the value of
lim
x
→
0
sin
2
x
sin
3
x
\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}
x
→
0
lim
sin
3
x
sin
2
x
.
Trigonometric Limits
Practice: Trig Limits
Evaluate
lim
x
→
0
2
sin
5
x
x
2
\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}
x
→
0
lim
x
2
2
sin
5
x
.
Practice: Using Known Trig Limits Indeterminate Forms
Behaviour of Trigonometric Functions: Evaluate
lim
x
→
0
sin
7
x
4
x
\lim_{x\to0}\limits\dfrac{\sin 7x}{4x}
x
→
0
lim
4
x
sin
7
x
Limits: Trigonometric Functions
Evaluate
lim
x
→
3
x
2
−
2
x
−
3
π
sin
(
x
−
3
)
\displaystyle\lim_{x\to 3}\dfrac{x^2-2x-3}{\pi\sin(x-3)}
x
→
3
lim
π
sin
(
x
−
3
)
x
2
−
2
x
−
3
Limits: Trigonometric Functions
Evaluate
lim
x
→
0
tan
2
x
4
x
(
x
2
−
1
)
\displaystyle\lim_{x\to 0}\dfrac{\tan 2x}{4x(x^2-1)}
x
→
0
lim
4
x
(
x
2
−
1
)
tan
2
x
Evaluate
lim
x
→
0
sin
7
x
4
x
\displaystyle \lim_{x\rightarrow 0}\frac{\sin{7x}}{4x}
x
→
0
lim
4
x
sin
7
x
.
Fundamental trigonometric limits
Evaluate
lim
x
→
0
tan
x
−
sin
x
x
cos
x
\lim\limits_{x\to0}\frac{\tan x-\sin x}{x\cos x}
x
→
0
lim
x
c
o
s
x
t
a
n
x
−
s
i
n
x
.
🌶️
TOUGH!
Evaluate
lim
x
→
π
sin
x
x
−
π
\lim\limits_{x\rightarrow \pi}\frac{\sin x}{x-\pi}
x
→
π
lim
x
−
π
s
i
n
x
. Hint: use the trigonometric identity
sin
(
θ
+
π
)
=
−
sin
(
θ
)
\sin(\theta+\pi)=-\sin(\theta)
sin
(
θ
+
π
)
=
−
sin
(
θ
)
.
Determine
lim
x
→
0
x
2
tan
2
3
x
\lim\limits_{x\rightarrow 0}\frac{x^2}{\tan^23x}
x
→
0
lim
t
a
n
2
3
x
x
2
.
Evaluate the following limit.
lim
x
→
0
tan
(
a
x
)
sin
x
\displaystyle \lim_{x\rightarrow 0}\frac{\tan(ax)}{\sin x}
x
→
0
lim
sin
x
tan
(
a
x
)
Consider
f
(
x
)
=
sin
2
x
x
\displaystyle f(x)=\frac{\sin{2x}}{x}
f
(
x
)
=
x
sin
2
x
. Does
lim
x
→
0
sin
2
x
x
\displaystyle \lim_{x\rightarrow 0}\frac{\sin{2x}}{x}
x
→
0
lim
x
sin
2
x
exist? Is
f
f
f
continuous at
x
=
0
x=0
x
=
0
?
Practice: Fundamental trigonometric limits
Evaluate
lim
x
→
1
−
sin
1
−
x
1
−
x
2
\lim\limits_{x\rightarrow1^-}\frac{\sin\sqrt{1-x}}{\sqrt{1-x^2}}
x
→
1
−
lim
1
−
x
2
s
i
n
1
−
x
Practice: Fundamental trigonometric limits
Part a)
Evaluate
lim
x
→
0
sin
(
2
π
x
)
sin
(
3
π
x
)
\lim\limits_{x\rightarrow0}\frac{\sin(2\pi x)}{\sin(3\pi x)}
x
→
0
lim
s
i
n
(
3
π
x
)
s
i
n
(
2
π
x
)
Part b)
Fundamental trigonometric limits
Determine
lim
x
→
0
x
2
tan
2
2
x
\lim\limits_{x\rightarrow0}\frac{x^2}{\tan^2\ 2x}
x
→
0
lim
t
a
n
2
2
x
x
2
.
Trig Limits
Find the value of
lim
x
→
0
sin
2
x
sin
3
x
\displaystyle \lim_{x\rightarrow0}\frac{\sin2x}{\sin3x}
x
→
0
lim
sin
3
x
sin
2
x
.
Trigonometric Limits
Practice: Trig Limits
Evaluate
lim
x
→
0
2
sin
5
x
x
2
\displaystyle\lim_{x\to0}\ \frac{2\sin5x}{x^2}
x
→
0
lim
x
2
2
sin
5
x
.
Fundamental Trig Limit
Evaluate the following limit:
lim
x
→
0
sin
(
3
x
)
cos
(
2
x
)
sin
(
5
x
)
\lim_{x\rightarrow0}\frac{\sin(3x)\cos(2x)}{\sin(5x)}
x
→
0
lim
sin
(
5
x
)
sin
(
3
x
)
cos
(
2
x
)
Each question is worth 3 marks.
Limits and Continuity
Evaluate the following limits:
a)
lim
x
→
4
x
−
4
x
−
4
\displaystyle \lim_{x \rightarrow 4} \frac{\sqrt{x} - \sqrt{4}}{x - 4}
x
→
4
lim
x
−
4
x
−
4
b)
lim
x
→
∞
x
2
−
5
x
+
13
4
x
4
+
3
x
−
21
\displaystyle \lim_{x \rightarrow \infty} \frac{x^2 - 5x + 13}{\sqrt{4x^4 +3x - 21}}
x
→
∞
lim
4
x
4
+
3
x
−
21
x
2
−
5
x
+
13