# 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.

Z =

Take the log to bring the power term in the product so that we can solve it more easily.

Taking log both sides-

log Z =

log Z =

{ log am = m log a}

using algebra of limits-

log Z =

log Z =

log Z =

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

log Z =

Let (x-1)/2 = y

As x1 y0

Z can be rewritten as

Log Z =

log Z =

Use the formula -

log Z =

log Z =

Z =

Hence,

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