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Solved Examples on Limits

Illustration 1: Find the following limit(1984)

limn→∞ [1/(1-n2) + 2/(1-n2) + … + n/(1-n2)]

Solution:limn→∞ [1/(1-n2) + 2/(1-n2) + … + n/(1-n2)]

 = limn→∞[1+2+3+…. +n]/ (1-n2)

             = limn→∞n(n+1)/ 2(1-n)(1+n)

             = limn→∞ n/ 2(1-n)

             = -1/2

Illustration 2: limx0 sin (π cos2x) / x2equals(2001)

1. –π                                                                   2. Π

3. π/2                                                                 4. 1

Solution: We need to compute the following expression

limx0 sin (π cos2x) / x2

= limx0 sin (π - πsin2x) / x2

= limx0 sin (π sin2x) / π sin2x .π sin2x/πx2 . π

= 1.1.π

 =n 

Illustration 3: let f: R→R besuch that f(1) = 3and f’(1) = 6. The find the value of limx0 [f(1+x)/ f(1)]1/x. (2002)

Solution: Let y = [f(1+x)/ f(1)]1/x

So, log y = 1/x[log f(1+x)...

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