Q. 42

# 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 =

As, 1-cos x = 2sin2(x/2)

Z =

Z =

To get the desired form to apply the formula we need to divide numerator and denominator by x2.

Z =

Using algebra of limits, we have-

Z =

Use the formula: and

Z =

Z = 2 log e = 2

Hence,

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