Q. 55.0( 2 Votes )

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 ∞-∞,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.


As, Z =


Z =


Taking log both sides-


log Z =


log Z =


{ log am = m log a}


Now it gives us a form that can be reduced to


Adding and subtracting 1 to cos x to get the form-


log Z =


Dividing numerator and denominator by cos x + a sin x– 1 to match with form in formula


log Z =


using algebra of limits –


log Z =


A =


Let, cos x + asin x - 1 = y


As x0 y0


A =


Use the formula -


A = 1


Now, B =


cos x – 1 = -2sin2(x/2) and sin x = 2sin(x/2)cos(x/2)


B =


B =


B =


Use the formula -


B =


B = 1/a


Hence,


log Z =


loge Z = a


Z = ea = ea


Hence,



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