Q. 29

# Evaluate the following limits:

As we need to find

We can directly find the limiting value of a function by putting the value of the variable at which the limiting value is asked if it does not take any indeterminate form (0/0 or ∞/∞ or ∞-∞, .. etc.)

Let Z =

We need to take steps to remove this form so that we can get a finite value.

TIP: Most of the problems of logarithmic and exponential limits are solved using the formula and

It also involves a trigonometric term, so there is a possibility of application of Sandwich theorem-

As Z =

To apply the formula we need to get the form as present in the formula. So we proceed as follows-

Z =

Multiplying numerator and denominator by √(1+cos x)

Z =

Using (a+b)(a-b) = a2-b2

Z =

√(1-cos2x) = sin x

Z =

{using algebra of limits}

Z =

Dividing numerator and denominator by x-

Z =

Z =

Use the formula: and

Z =

{ log e = 1}

Hence,

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