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Title: An important limit in calculus
Description: An important limit in Calculus explained analytically and gramatically with graph to create visual learning. All relevant formulas included.
Description: An important limit in Calculus explained analytically and gramatically with graph to create visual learning. All relevant formulas included.
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Limit
shahbaz ahmed
August 2024
1
Reminisce
THE SQUEEZE (SANDWICH) THEOREM
Let g(x) and h(x) be functions such that g(x) ≤ f (x) ≤ h(x) on a
punctured neighborhood of a
...
The line segment AB is a tangent to the circle of radius a at the
point B such that OAB is a right angled triangle
...
Where 0 < x <
π
2
OB=OC=radii of the circle=a
CD is an altitude on OB
...
AB
OB
=
AB
a
= tan x
AB = a tan x
2
Similarly
Area of the circle of radius a with 2π Radians=πa2
2
Area of the sector of a circle of radius a with 1 Radians= πa
2π =
a2
2
2
Area of the sector of a circle of radius a with x Radians= a2 x
Also, in the right triangle OCD
CD
OC
=
CD
a
= sin x
CD = a sin x
Area of the triangle OAB= 21 (OB)(AB) = 12 a(a tan x) = 21 a2 tan x
Area of the triangle OCB= 12 (OB)(CD) = 21 a(a sin x) = 12 a2 sin x
Reference to the figure
⌢
Area of the triangle OCB ≤ Area of the sector of the circle OCB ≤
Area of the triangle OAB
1 2
2 a sin x
2
≤ a2 x ≤ 21 a2 tan x,
0
sin x ≤x ≤ tan x
1 ≤ sinx x ≤
sin x
cos x
1 ≤ sinx x ≤
1
cos x
×
1
sin x
1 ≤ limx−→0 sinx x ≤ limx−→0 cos1 x
Now
limx−→0 cos1 x = 1, since cos 0 = 1
3
π
2
Hence
1 ≤ limx−→0 sinx x ≤ 1
BY THE SQUEEZE (SANDWICH) THEOREM
x
=1
x−→0 sin x
lim
Similarly
sin x
=1
x−→0 x
lim
3
Formulas used
cos 2x = cos2 x − sin2 x
cos2 x = 1 − sin2 x
sin2 x = 1 − cos2 x
...
√
2θ = x
θ −→ 0+ ⇐⇒ x −→ 0+ , x > 0, θ > 0
√
sin x
sin 2θ
=1
lim+ √
= lim+
θ−→0
x−→0
x
2θ
As
√
2θ is undefined for θ < 0, so
limx−→o−
sin x
x
does not exist
...
The graph below
as x get closer to zero
Title: An important limit in calculus
Description: An important limit in Calculus explained analytically and gramatically with graph to create visual learning. All relevant formulas included.
Description: An important limit in Calculus explained analytically and gramatically with graph to create visual learning. All relevant formulas included.