HOW TO EVALUATE LIMITS WITH TRIG FUNCTIONS

Question 1 :

Evaluate the following limit 

lim  _{x -> 0} sin³ (x/2)/ x³

Solution :

lim  _{x -> 0} sin³ (x/2)/ x³  =  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 ᵦx

Solution :

=  lim  _{x -> 0} sin αx / sin ᵦx

=  (lim  _{x -> 0} sin αx) (αx/αx) / (lim  _{x -> 0} sin ᵦx)(ᵦx/ᵦx)

= (αx/ᵦx) (lim  _{x -> 0} sin αx/αx) / (lim  _{x -> 0} sin ᵦx/ᵦx)

=  (α/ᵦ) (1)

=  α/ᵦ

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} (α^n/α^m) lim  _{α -> 0} (sin αⁿ/αⁿ) /lim  _{α -> 0} (sin α /α)^m

=   lim  _{α -> 0} (α^n/α^m) 

=  lim  _{α -> 0} α^n-m

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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