The Fundamental Trig Limit
The Fundamental Trig Limit
This is it:
The Fundamental Trig Limit
This is it:
limx→0
sin x
x= 1
The Fundamental Trig Limit
This is it:
limx→0
sin x
x= 1
This is proved using the squeeze theorem!
Example 1
Example 1
limx→0
sin 4x
x
Example 1
limx→0
sin 4x
x
Here the trick is to multiply and divide by 4:
Example 1
limx→0
sin 4x
x
Here the trick is to multiply and divide by 4:
limx→0
sin 4x
x=
Example 1
limx→0
sin 4x
x
Here the trick is to multiply and divide by 4:
limx→0
sin 4x
x= lim
x→0
Example 1
limx→0
sin 4x
x
Here the trick is to multiply and divide by 4:
limx→0
sin 4x
x= lim
x→0
4 sin 4x
4x
Example 1
limx→0
sin 4x
x
Here the trick is to multiply and divide by 4:
limx→0
sin 4x
x= lim
x→0
4 sin 4x
4x
= 4 limx→0
sin 4x
4x
Example 1
limx→0
sin 4x
x
Here the trick is to multiply and divide by 4:
limx→0
sin 4x
x= lim
x→0
4 sin 4x
4x
= 4 limx→0�
���>
1sin 4x
4x
Example 1
limx→0
sin 4x
x
Here the trick is to multiply and divide by 4:
limx→0
sin 4x
x= lim
x→0
4 sin 4x
4x
= 4 limx→0
sin 4x
4x
Example 1
limx→0
sin 4x
x
Here the trick is to multiply and divide by 4:
limx→0
sin 4x
x= lim
x→0
4 sin 4x
4x
= 4 limx→0
sin 4x
4x= 4.1
Example 1
limx→0
sin 4x
x
Here the trick is to multiply and divide by 4:
limx→0
sin 4x
x= lim
x→0
4 sin 4x
4x
= 4 limx→0
sin 4x
4x= 4.1 = 4
Example 1
limx→0
sin 4x
x
Here the trick is to multiply and divide by 4:
limx→0
sin 4x
x= lim
x→0
4 sin 4x
4x
= 4 limx→0
sin 4x
4x= 4.1 = 4
Example 2
Example 2
limx→0
tan x
x
Example 2
limx→0
tan x
x
Here we use a basic trig identity:
Example 2
limx→0
tan x
x
Here we use a basic trig identity:
limx→0
tan x
x=
Example 2
limx→0
tan x
x
Here we use a basic trig identity:
limx→0
tan x
x= lim
x→0
Example 2
limx→0
tan x
x
Here we use a basic trig identity:
limx→0
tan x
x= lim
x→0
sin xcos x
x
Example 2
limx→0
tan x
x
Here we use a basic trig identity:
limx→0
tan x
x= lim
x→0
sin xcos x
x
= limx→0
sin x
x.
1
cos x
Example 2
limx→0
tan x
x
Here we use a basic trig identity:
limx→0
tan x
x= lim
x→0
sin xcos x
x
= limx→0�
���
1sin x
x.
1
cos x
Example 2
limx→0
tan x
x
Here we use a basic trig identity:
limx→0
tan x
x= lim
x→0
sin xcos x
x
= limx→0
sin x
x.����
11
cos x
Example 2
limx→0
tan x
x
Here we use a basic trig identity:
limx→0
tan x
x= lim
x→0
sin xcos x
x
= limx→0
sin x
x.
1
cos x=
Example 2
limx→0
tan x
x
Here we use a basic trig identity:
limx→0
tan x
x= lim
x→0
sin xcos x
x
= limx→0
sin x
x.
1
cos x= 1
Example 2
limx→0
tan x
x
Here we use a basic trig identity:
limx→0
tan x
x= lim
x→0
sin xcos x
x
= limx→0
sin x
x.
1
cos x= 1
Example 3
Example 3
limx→0
1 − cos x
x
Example 3
limx→0
1 − cos x
x
In this case a somewhat more elaborate trig identity will be useful:
Example 3
limx→0
1 − cos x
x
In this case a somewhat more elaborate trig identity will be useful:
sinx
2=
√1 − cos x
2
Example 3
limx→0
1 − cos x
x
In this case a somewhat more elaborate trig identity will be useful:
sinx
2=
√1 − cos x
2
Squaring both sides and solving for 1 − cos x we get:
Example 3
limx→0
1 − cos x
x
In this case a somewhat more elaborate trig identity will be useful:
sinx
2=
√1 − cos x
2
Squaring both sides and solving for 1 − cos x we get:
1 − cos x = 2 sin2 x
2
Example 3
Let’s replace that in our limit:
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x=
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
2 sin2 x2
x
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
2 sin2 x2
x
= 2 limx→0
sin x2
x. sin
x
2
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
2 sin2 x2
x
= 2 limx→0
sin x2
x. sin
x
2
Now the trick is to multiply and divide by 2 in the first factor:
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
2 sin2 x2
x
= 2 limx→0
sin x2
x. sin
x
2
Now the trick is to multiply and divide by 2 in the first factor:
= 2 limx→0
sin x2
x2 .2
. sinx
2
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
2 sin2 x2
x
= 2 limx→0
sin x2
x. sin
x
2
Now the trick is to multiply and divide by 2 in the first factor:
= �2 limx→0
sin x2
x2 .�2
. sinx
2
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
2 sin2 x2
x
= 2 limx→0
sin x2
x. sin
x
2
Now the trick is to multiply and divide by 2 in the first factor:
= �2 limx→0
sin x2
x2 .�2
. sinx
2
= limx→0
sin x2
x2
. sinx
2
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
2 sin2 x2
x
= 2 limx→0
sin x2
x. sin
x
2
Now the trick is to multiply and divide by 2 in the first factor:
= �2 limx→0
sin x2
x2 .�2
. sinx
2
= limx→0����7
1sin x
2x2
. sinx
2
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
2 sin2 x2
x
= 2 limx→0
sin x2
x. sin
x
2
Now the trick is to multiply and divide by 2 in the first factor:
= �2 limx→0
sin x2
x2 .�2
. sinx
2
= limx→0
sin x2
x2
.����
0
sinx
2
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
2 sin2 x2
x
= 2 limx→0
sin x2
x. sin
x
2
Now the trick is to multiply and divide by 2 in the first factor:
= �2 limx→0
sin x2
x2 .�2
. sinx
2
= limx→0
sin x2
x2
. sinx
2
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
2 sin2 x2
x
= 2 limx→0
sin x2
x. sin
x
2
Now the trick is to multiply and divide by 2 in the first factor:
= �2 limx→0
sin x2
x2 .�2
. sinx
2
= limx→0
sin x2
x2
. sinx
2= 1.0
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
2 sin2 x2
x
= 2 limx→0
sin x2
x. sin
x
2
Now the trick is to multiply and divide by 2 in the first factor:
= �2 limx→0
sin x2
x2 .�2
. sinx
2
= limx→0
sin x2
x2
. sinx
2= 1.0 = 0
Example 3
Let’s replace that in our limit:
limx→0
1 − cos x
x= lim
x→0
2 sin2 x2
x
= 2 limx→0
sin x2
x. sin
x
2
Now the trick is to multiply and divide by 2 in the first factor:
= �2 limx→0
sin x2
x2 .�2
. sinx
2
= limx→0
sin x2
x2
. sinx
2= 1.0 = 0
Recommended
