Q. 45.0( 2 Votes )

# 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 ∞-∞,1 .. etc.)

Let Z =

As it is taking indeterminate form-

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

As, Z =

Z =

Taking log both sides-

log Z =

log Z =

{ log am = m log a}

Now it gives us a form that can be reduced to

log Z =

{adding and subtracting 1 to cos x to get the form}

Dividing numerator and denominator by cos x + sin x– 1 to match with form in formula

log Z =

using algebra of limits –

log Z =

A =

Let, cos x + sin x - 1 = y

As x0 y0

A =

Use the formula -

A = 1

Now, B =

cos x – 1 = -2sin2(x/2) and sin x = 2sin(x/2)cos(x/2)

B =

B =

B =

Use the formula -

B =

B = 1

Hence,

log Z =

loge Z = 1

Z = e1 = e

Hence,

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