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 a^m = m log a}

Now it gives us a form that can be reduced to

⇒ log Z =

Dividing numerator and denominator by to get the desired form and using algebra of limits we have-

log Z =

if we assume then as x→a ⇒ y→ 0

⇒ log Z =

Use the formula-

∴ log Z =

⇒ log Z =

Now it gives us a form that can be reduced to

Try to use it. We are basically proceeding with a hit and trial attempt.

⇒ log Z =

∵ sin (A+B) = sin A cos B + cos A sin B

⇒ log Z =

⇒ log Z=

⇒ log Z =

⇒ log Z =

Use the formula-

⇒ log Z = cot a – 0

∴ log Z = cot a

∴ Z = e^{cot a}

Hence,