By Gökhan Bilhan
1
Calculus 1
(Week 6)-Limits of Exponential, Logarithmic and Trigonometric Functions
Example
limx→1 ex
x2 −x
2 −x
limx→1 e x−1
Example
1
limx→−∞ ex
limx→0− e x
By looking at the graph of
limx→∞ lnx =
and
Let's look at the graph of
Then obviously
limx→0
Examples
1-)
limx→0
sin2 x
=
x2
lnx,
sinx
=
x
we may say that
limx→0 lnx =
By Gökhan Bilhan
2
2-)
sin2 x
limx→0
=
x
3-)
limh→0
cosh − 1
=
h
Exercises
1. Find
a∈R
so that the following limit exists.
limx→0 (
x2 + ax
)
1 − cosx
2.
limx→0 (
1 − cos(tanx)
)
tanx
3.
limx→0 (
sin7x
)
tan7x
By Gökhan Bilhan
4.
sin2 x
limx→0 (
)
1 − secx
5.
limx→0
3
sin(sin(sinx))
sin(sinx)
Theorem(The Sandwich Theorem) Suppose that g(x) ≤ f (x) ≤ h(x) for all x in some open interc, except
limx→c f (x) = L.
val containing
then
possibly at
x=c
itself. Suppose also that
limx→c g(x) = limx→c h(x) = L,
Theorem The trigonometric functions y = sinx and y = cosx are continuous everywhere.
Denition A function f (x) is called bounded near x0 , if |f (x)| ≤ M
x
near to
(M is real number)for every
x0
Theorem If f (x) is bounded near a and if limx→a g(x) = 0 then limx→a f (x).g(x) = 0
Example limx→0 (x.sin
1
)
x
Example limx→0 (sinx.sin
(Note that
1
)
x
|sinx| ≤ 1
and
|cosx| ≤ 1
for all
x.)
By Gökhan Bilhan
4
Example limx→0 sinx(cotx).sinx
Example limx→0 tanx.sin
1
x
Exercises
π
limx→π 2 2 2 =?
x −π
cos
1.
2.
limx→1
sinπx
=?
x2 − 1
3.
limx→1
1
πx
.cos
=?
1−x
2
4.
limx→0
sin(sinx)
=?
x
By Gökhan Bilhan
5
5.
limx→∞ x2 − x =
6.
limx→∞
x2 + x
=
3−x
By Gökhan Bilhan
6
(Week 5)-Trigonometric and Other Limits(Exercises)
Exercises
1.
lim
π(
x→
3
A)2
2.
3.
4.
lim
√
√
B)
3
π(
x→
6
3−1
C)0
D)−
√
E)−2
3
√
3−1
B)
limx→∞
sin3x
=?
3x
A)0
B)1
C)
C)3
√
1
(1 − 3)
2
D)4
D)
√
A)sin
9
limx→0
tanx − sinx
=?
sin3 x
1
3
√
3
(1 + 3)
π
E)No limit
√
limx→−π sin 10 + cosx =?
A) 2
√
sinx + cosx
) =?
π
−x
3
A)0
√
5.
2sinx − tanx
) =?
cosx
B)1
B)cos
C)0
9
C)0
3
D) 2
D)1
E)2
E)sin9
E)
π
3
By Gökhan Bilhan
6.
π
cos( x)
2
limx→1
=?
sinπx
A)
7.
7
− 12
lima→x
A)1
B)− 4
1
C)0
1
D) 4
1
E) 2
x2 − a2
=?
sin(2x − 2a)
B)x
C)a
D)2a
E)
x
2
Answers:
1-) C , 2-)D , 3-)A , 4-)A , 5-)A , 6-)E , 7-)B

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