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Limit and Continuity of Trigonometric Functions

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Calculus and Analytical Geometry
MTH101
LECTUER ­ 13
Limit and Continuity of
Trigonometric Functions
B(1,0)
It is evident that as h approaches 0, the
coordinate of P approach the corresponding
coordinate of B.
Continuity of Sine and Cosine
function
But by definition we know
Here is the graph of Sinx and Cosx
Remind the definition of continuity we have
1
Hence we have the following theorem
0
-1
1
0
-1
Mth101
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Calculus and Analytical Geometry
Limit and Continuity of
Trigonometric Functions
Proof
Similarly, since
Obtaining Limits by Squeezing
Mth101
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Calculus and Analytical Geometry
Now
Consider the graph of
o
Hence proved.
And
Limit and Continuity of
Trigonometric Functions
And the graph of
o
Here the problem is:
Before proving next theorem, let us see
the following formula.
Here we cannot write these functions
in some other form by using algebraic
Manipulation to solve this problem. So here
we will use some other method. One such
method is to obtain by following theorem;
Mth101
Page 40
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Calculus and Analytical Geometry
Example: Use the squeezing theorem
to evaluate
Solution
Limit and Continuity of
Trigonometric Functions
So, the above inequality becomes
Mth101
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Calculus and Analytical Geometry
Mth101
Page 42