TRIGONOMETRIC LIMITS PROBLEMS AND SOLUTIONS

Problem 1 :

Evaluate the following limit 

lim x-> 0  (1 + sin x)2 cosec x

Solution :

  =  lim x-> 0  (1 + sin x)2 cosec x

Let y = sin x

If x -> 0, then y -> 0

cosec x  =  1/ sin x  =  1/y

lim x-> 0  (1 + sin x)2 cosec x  =  lim x-> 0  (1 + y)2/y

 lim x-> 0  (1 + x)1/x  =  e

  =  e2

Problem 2 :

Evaluate the following limit 

 lim x-> 0 (√2 - √(1 + cos x)) / sin2 x

Solution :

  =  lim x-> 0  (√2 - √(1 + cos x)) / sin2 x

By applying the limit value directly in the given question, we get 0/0. That is indeterminate form.

So let us rationalize the numerator,

  =  limx->0((√2-√(1+cosx))/sin2x)[(√2+√(1+cosx))/(√2+√(1+ cosx))]

  =  limx->0((2-(1+cosx))/sin2x(√2+√(1+ cosx))

  =  limx->0 (1-cosx)/[1+cosx)(1-cosx)(√2+√(1+ cosx))]

  =  limx->0 1/[(1+cosx)(√2+√(1+ cosx))]

  =  1/[2(√2+√(1+1)]

  =  1/4√2

Hence the value of  lim x-> 0 (√2 - √(1 + cos x)) / sin2 x is 1/4√2.

Problem 3 :

Evaluate the following limit 

 lim x-> 0 (√(1+sinx) - √(1-sinx)) / tan x

Solution :

  =   lim x-> 0 (√(1+sinx) - √(1-sinx)) / tan x

By rationalizing the numerator, we get

  =   lim x-> 0 (1+sinx) - (1-sinx)/tan x(√(1+sinx)+√(1-sinx))

  =   lim x-> 0 2sinx/tan x(√(1+sinx)+√(1-sinx))

tan x =  sin x/cos x

  =   lim x-> 0 2 cos x/(√(1+sinx)+√(1-sinx))

By applying the limit, we get

  =   2 /2

  =  1

Hence the value of  lim x-> 0 (√(1+sinx) - √(1-sinx)) / tan x is 1.

Problem 4 :

Evaluate the following limit

lim x-> ∞  {(x2 – 2x + 1)/(x2-4x+2))x

Solution :

  =  lim x-> ∞  {(x2 – 2x + 1)/(x2-4x+2))x

By applying the limit value directly in the given question, we get indeterminant form.

  =  lim x-> ∞  {(x2 – 2x + 1)/(x2-4x+2))x

  =  lim x-> ∞  {1 + ((2x - 1)/(x2-4x+2))x}

  =  lim x-> ∞  {1 + ((2x - 1)/(x2-4x+2))

Let y = (2x- 1)/(x2-4x+2)

If x -> , then y =  xx(2- 1/x)/x2(1-4/x+2/x2), y -> 2

  =  lim y-> 0 {(1 + y)1/y}y

  =  ey

By applying the value of y, we get

  =  e2

Hence the value of lim x-> ∞  {(x2 – 2x + 1)/(x2-4x+2))x is e2

Problem 5 :

Evaluate the following limit

lim x-> 0  (ex - e-x) / sin x

Solution :

  =  lim x-> 0  (ex - e-x) / sin x

  =  lim x-> 0  (ex - (1/ex)) / sin x

  =  lim x-> 0  ((ex)2 - 1)/ex sin x

  =  lim x-> 0  (e2x - 1)/ex sin x

Now we are going to multiply numerator by 2x/2x, sin x by (x/x)

  =  lim x-> 0  (e2x - 1)(2x/2x)/ex sin x (x/x)

  =  lim x-> 0  ((e2x - 1)/2x)(2x/x)/(ex (sin x/x))

  =  2lim x-> 0 ((e2x - 1)/2x)/lim x-> 0(ex lim x-> 0(sin x/x))

  =  2(1)/1(1)

  =  2

Hence the value of lim x-> 0  (ex - e-x) / sin x is 2.

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