# LIMITS EXAMPLES AND SOLUTIONS

## About "Limits Examples and Solutions"

Limits Examples and Solutions :

Here we are going to see some example questions on evaluating limits.

To find the formulas please visit "Formulas in evaluating limits".

## Limits Examples and Solutions - Practice questions

Question 1 :

Evaluate the following limit

lim x-> 0  (1 – cos2x)/(x sin 2x)

Solution :

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

=  lim x-> 0  (1/2)(sinx)/(x cos x)

=  (1/2) lim x-> 0  (tanx/x)

=  (1/2) (1)

=  1/2

Question 2 :

Evaluate the following limit

lim x-> ∞  x (31/x + 1 – cos (1/x) – e1/x)

Solution :

=  lim x-> ∞  x (31/x + 1 – cos (1/x) – e1/x)

=  lim x-> ∞  (31/x + 1 – cos (1/x) – e1/x)/(1/x)

let y = 1/x

If x -> ∞, then y -> 0

=  lim  y-> 0   (3y + 1 – cos y – ey)/y

=  lim  y-> 0 (3y + 1 – cos y – ey + 1 - 1)/y

=  lim  y-> 0  [(3y - 1) + (1 – cos y) - (e- 1)]/y

=  [lim y-> 0 (3y-1)/y]+[lim  y-> 0 (1–cos y)/y]-[lim  y-> 0 (ey-1)]/y]

=  log 3 + 0 - 1

=  log 3 - 1

Hence the value of lim x-> ∞  x (31/x + 1 – cos (1/x) – e1/x) is log 3 - 1.

Question 3 :

Evaluate the following limit

lim x-> ∞  {x [log (x + a) – log x] }

Solution :

=  lim x-> ∞  {x [log (x + a) – log x] }

=  lim x-> ∞  {x [log (x + a)/x] }

=  lim x-> ∞  (log (1 + a/x))/(1/x)

=  lim x-> ∞  (a/a)(log (1 + a/x))/(1/x)

=  lim x-> ∞  a(log (1 + a/x))/(a/x)

Let y = a/x

If x -> ∞, then y -> 0

=  lim y -> 0  a(log (1 + y )/y)

=  a

Question 4 :

Evaluate the following limit

lim x-> π  sin 3x/sin 2x

Solution :

=  lim x-> π  sin 3x/sin 2x

By applying the limit in the given question, we get 0/0.

=  lim x-> π  sin (2x + x)/sin 2x

=  lim x-> π  (sin 2x cos x + cos 2x sin x)/sin 2x

=  lim x-> π [(sin 2x cos x)/sin 2x + (cos 2x sin x)/sin 2x]

Applying the formula for sin 2x  =  2 sin x cos x

=  lim x-> π cos x + lim x-> π (cos 2x sin x)/(2 sin x cos x)

=  cos π + cos 2π/2cos π

=  -1 - (1/2)

=  -3/2

Hence the value of lim x-> π  sin 3x/sin 2x is -3/2.

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