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 am = m log a}
⇒ log Z =
{using algebra of limits}
Still, if we put x = ∞ we get an indeterminate form,
Take highest power of x common and try to bring x in denominator of a term so that if we put x = ∞ term reduces to 0.
∴ log Z =
⇒ log Z =
⇒ log Z =
⇒ log Z =
{∵ log (3/4) is a negative value as 3/4<1}
∴ Loge Z = -∞
⇒ Z = e-∞ = 0
Hence,