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 am = 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 = ecot a
Hence,