SOLVING LIMITS EXAMPLES

Example 1 :

Evaluate the following limit :

lim _{x-> 0}  (e^ax - e^bx)/x

Solution :

  =  lim _{x-> 0}  (e^ax - e^bx)/x

  =  lim _{x-> 0}  (e^ax - e^bx + 1 - 1)/x

  =  lim _{x-> 0}  [(e^ax - 1) - (e^bx - 1)]/x

  =  lim _{x-> 0}  [(e^ax - 1)/x] - lim _{x-> 0}[(e^bx - 1)/x]

  =  lim _{x-> 0}  [a(e^ax - 1)/ax] - lim _{x-> 0}[b(e^bx - 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))² lim _{x-> 0}(1/cos x)

=  1/2

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