HOW TO EVALUATE LIMITS WITH TRIG FUNCTIONS

lim θ-->0 (sinθ/θ) = 1

limθ-->0 (1 - cosθ)/θ = 0

limθ-->0 (tanθ)/θ = 1

limx-->0 (sin-1x)/x = 1

limx-->0 (tan-1x)/x = 1

limx-->a [sin(x - a)]/(x - a) = 1

limx-->a [tan(x - a)]/(x - a) = 1 

Question 1 :

Evaluate the following limit 

lim  x -> 0 sin3 (x/2)/ x3

Solution :

lim  x -> 0 sin3 (x/2)/ x=  lim  x -> 0 (sin (x/2))3/ x3

In order to match the given question with the formula, let us multiply and divide by 1/8

 =  lim  x -> 0 (sin (x/2))3(1/8)/ x3(1/8)

 =  (1/8) lim  x -> 0 (sin (x/2))3/ (x/2)3

 =  (1/8) lim  x -> 0 [sin (x/2) / (x/2)]3

 =  1/8 (1)

  =  1/8

Hence the value of lim  x -> 0 sin3 (x/2)/ x3 is 1/8.

Question 2 :

Evaluate the following limit 

lim  x -> 0 sin αx / sin x

Solution :

=  lim  x -> 0 sin αx / sin ᵦx

=  (lim  x -> 0 sin αx) (αx/αx) / (lim  x -> 0 sin x)(x/x)

= (αx/x) (lim  x -> 0 sin αx/αx) / (lim  x -> 0 sin x/x)

=  (α/ᵦ) (1)

α/

Hence the value of lim  x -> 0 sin αx / sin ᵦx is α/ᵦ.

Question 2 :

Evaluate the following limit 

lim  x -> 0 tan 2x / sin 5x

Solution :

=  lim  x -> 0 tan 2x / sin 5x

=  lim  x -> 0 tan 2x  (2x/2x) / lim  x -> 0 sin 5x (5x/5x)

=  (2x/5x) [(lim  x -> 0 tan 2x/2x) / (lim  x -> 0 sin 5x/5x)

=  (2/5) (1/1)

=  2/5

Hence the value of lim  x -> 0 tan 2x / sin 5x is 2/5.

Question 3 :

Evaluate the following limit 

lim  α -> 0 (sin αn)/ (sin α)m

Solution :

=  lim  α -> 0 (sin αn⋅ (αn/αn/ (sin α  (α/α))m

 lim  α -> 0 (αn/αmlim  α -> 0 (sin αn/αn/lim  α -> 0 (sin α /α)m

=   lim  α -> 0 (αn/αm

=  lim  α -> 0 αn-m

If n = m

=  lim  α -> 0 αn-n

  =  0n - n

  =  1

If m > n

=  lim  α -> 0 αn-m

  =  0n - m

  =  0negative value

=  0  

If m < n

=  lim  α -> 0 αn-m

  =  0n - m

  =  0positive value

 

Question 4 :

Evaluate the following limit 

lim  x -> 0 (sin (a + x) - sin (a - x))/ x

Solution :

=  lim  x -> 0 (sin (a + x) - sin (a - x)) / x

sin C - sin D  =  2 cos ((C + D)/2) sin ((C - D)/2)

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

=  lim  x -> 0 (2 cos a sin x)/x

 2 cos a  lim  x -> 0 (sin x/x)

 2 cos a (1)

=  2 cos a

Hence the value lim  x -> 0 (sin (a + x) - sin (a - x))/ x is 2 cos a.

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