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 ∞-∞, .. 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 =


As, 1-cos x = 2sin2(x/2)


Z =


Z =


To get the desired form to apply the formula we need to divide numerator and denominator by x2.


Z =


Using algebra of limits, we have-


Z =


Use the formula: and


Z =


Z = 2 log 2


Hence,



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