Limits of Trigonometric Functions

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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: