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 term in the product so that we can solve it more easily.

Taking log both sides-

log Z =

⇒ log Z =

{∵ log a^m = m log a}

⇒ log Z =

{using algebra of limits}

Still, if we put x = ∞ we get an indeterminate form,

Take the highest power of x common and try to bring x in the denominator of a term so that if we put x = ∞ term reduces to 0.

∴ log Z =

⇒ log Z =

⇒ log Z =

⇒ log Z =

∴ Log_e Z =

⇒ Z = 1/2

Hence,