Q. 29

Evaluate the following limits:


Answer :

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 ∞-∞, .. etc.)


Let Z =


We need to take steps to remove this form so that we can get a finite value.


TIP: Most of the problems of logarithmic and exponential limits are solved using the formula and


It also involves a trigonometric term, so there is a possibility of application of Sandwich theorem-


As Z =


To apply the formula we need to get the form as present in the formula. So we proceed as follows-


Z =


Multiplying numerator and denominator by √(1+cos x)


Z =


Using (a+b)(a-b) = a2-b2


Z =


√(1-cos2x) = sin x


Z =


{using algebra of limits}


Z =


Dividing numerator and denominator by x-


Z =


Z =


Use the formula: and


Z =


{ log e = 1}


Hence,



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