Functions Limits and Continuity


   
 
Limits (Contd....)
Limits of Trigonometric Functions
 
Before describing the limits of trigonometric functions, we state few theorems along with Sandwich theorem, which helps in calculating a variety of limits in subsequent chapters.
 
Theorem 2:
 
Let f and g be real valued functions defined on an interval containing c such that  exist. Then
 
 
The following statement is not true.
 
f(x) < g(x) for all x
 
 
Theorem 3:
 
If f is a function defined on an open interval containing c, then
 
 
 
Theorem 4 (Sandwich Theorem):
 
 
for all x in some open interval containing c and suppose
 
 
 
 
Since f is sandwiched between two functions g and h, the above theorem is known as sandwich theorem.
 
Theorem 5:
 
 
Proof:
 
Consider a circle with centre O and radius r.
 
 
Join AB. Let the tangent at B meet OA produced at P. Draw BN perpendicular to OA.
 
 
From ONB,
 
 
BN = r sin q
 
From OBP,
 
 
BP = r tan q
 
From the figure, we have
 
Area of triangle OAB < Area of sector OAB < Area of triangle OBP
 
 
 
 
 
 
 
 
 
Note 1:
 
 
Note 2:
 
 
 
= 1
 
Note 3:
 
 
Limits Involving Exponential Functions
 
Theorem 6:
 
 
Proof:
 
 
 
 
 
 
 
We know that,
 
 
 
 
 
 
Further, we have
 
 
 
Substituting this value in (2), we have
 
 
From (1) and (3), we have
 
 
 
it follows from the above inequation that
 
 
 
 
\ From equation(4), we get
 
 
 
(Note that -x > 0, so multiplying this in equation by -x, the inequality remains same)
 
 
Add 1 on both sides,
 
 
Taking the reciprocal, we have
 
 
Subtracting 1, we have
 
 
 
Diving by the negative number x, we get
 
 
 
 
 
Now x < 0, |x| = - x
 
 
 
From (5) and (6), the theorem is proved.
 
Theorem 7:
 
 
Proof:
 
 
From the above theorem, we have
 
 
 
 
 
 
= 1
 
\ Using Sandwich theorem, we get
 
 
Example:
 
 
Suggested answer:
 
 
 
 
 
 
=1
 
Theorem 8:
 
 
Proof:
 
 
 
 
 
 
 
Theorem 9:
 
 
Proof:
 
 
 
 
 
 
 
  
     
   
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