SOLVING LIMITS EXAMPLES

Example 1 :

Evaluate the following limit :

lim x-> 0  (eax - ebx)/x

Solution :

  =  lim x-> 0  (eax - ebx)/x

  =  lim x-> 0  (eax - ebx + 1 - 1)/x

  =  lim x-> 0  [(eax - 1) - (ebx - 1)]/x

  =  lim x-> 0  [(eax - 1)/x] - lim x-> 0[(ebx - 1)/x]

  =  lim x-> 0  [a(eax - 1)/ax] - lim x-> 0[b(ebx - 1)/bx]

  = a(1) - b(1)

=  a - b

Example 2 :

Evaluate the following limit :

lim x-> 0  sin x (1 - cos x)/x3

Solution :

  =  lim x-> 0  (sin x/x) ((1 - cos x)/x2)

  =  lim x-> 0  (sin x/x) lim x-> 0 ((1 - cos x)/x2)

=  1 lim x-> 0 (2 sin2 x/2)/x2

lim x-> 0 (2 sin2 x/2)/x2(4/4)

=  lim x-> 0 (2/4) [(sin x/2)/(x/2)]2

=  1/2

Example 3 :

Evaluate the following limit :

lim x-> 0  (tan x - sin x)/x3

Solution :

  =  lim x-> 0  (tan x - sin x)/x3

=  lim x-> 0  ((sin x/ cos x) - sin x)/x3

=  lim x-> 0  (sin x/x)((1 - cos x)/x2cos x)

=  lim x-> 0  (sin x/x)lim x-> 0((2 sin2x/2)/x2cos x)

=  (1) lim x-> 0((2 sin2x/2)/x2(4/4) cos x)

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

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

=  1/2

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