Solved Examples on Limits

Illustration 1: Find the following limit(1984)

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

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

 = lim_n→∞[1+2+3+…. +n]/ (1-n²)

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

             = lim_n→∞ n/ 2(1-n)

             = -1/2

Illustration 2: lim_x→0 sin (π cos²x) / x²equals(2001)

1. –π                                                                   2. Π

3. π/2                                                                 4. 1

Solution: We need to compute the following expression

lim_x→0 sin (π cos²x) / x²

= lim_x→0 sin (π - πsin²x) / x²

= lim_x→0 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 lim_x→0 [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) – log  f(1)]

So, lim_x→0 log y = lim_x→0[1/f(1+x) . f’(1+x)]

                           = f’(1)/ f(1)

                           = 6/3

log (lim_x→0 y) =2

lim_x→0 y = e²