Important Questions for Class 11 Chapter 13 - Limits and Derivatives

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Important questions based on NCERT syllabus for Class 11 Chapter 13 - Limits and Derivatives:

Question 1: Evaluate lim x-> 0 xtan4x/1-cos4x

Solution: Consider lim x-> 0 xtan4x/1-cos4x

= lim x->0 xtan4x/cos4x(2sin^2 2x)

= lim x->0 2xsin2x cos2x/cos4x(2sin^2 2x)

= lim x->0(cos 2x/cos 4x * 2x/sin2x * 1/2)

= 1/2 lim 2x->0 cos 2x / lim 4x->0 cos 4x * lim 2x->0 (2x/sin 2x)

= 1/2 * 1 = 1/2

Question 2: Find the indicated limit lim x->0 √(3-x+x^2-3)/x

Solution: lim x->0 √(3-x+x^2-3)/x

= lim x->0 √(3-x+x^2-3)/x * √(3-x+x^2+3)/√(3-x+x^2+3)

= lim x->0 (3-x+x^2-3)/ x√(3-x+x^2+3)

= lim x->0 -x+x^2/ x√(3-x+x^2+3)

= lim x->0 (-1+x)/ √(3-x+x^2+3)

= -1/√(3+3)

= 1/√(6)

Question 3: Compute lim x->0 (a^x - b^x)/x, a,b>0. and hence evaluate lim x->0 (8^x - 9^x)/x

Solution: lim x->0 (a^x - b^x)/x

= lim x->0 ((a^x-1) - (b^x-1))/x

= lim x->0 ((a^x-1)/x - lim x->0 (b^x-1)/x

= log a - log b

= log (a/b)........(1)

Hence to evaluate lim x->0 (8^x - 9^x)/x we the equation (1)

lim x->0 (8^x - 9^x)/x = log(8/9)