HOW TO EVALUATE LIMITS WITH TRIG FUNCTIONS

About "How to Evaluate Limits With Trig Functions"

How to Evaluate Limits With Trig Functions :

Here we are going to see some practice questions on evaluating limits with trig functions.

Before going to see example problems, let us see the formulas based on trigonometric functions used in evaluating limits.

(1)  lim θ -> 0 sin θ / θ  =  1

(2)  lim θ -> 0 (1 - cos θ) / θ  =  0

(3)  lim θ -> 0 tan θ / θ  =  1

(4)  lim x -> 0 sin-1x/x  =  1

(5) lim x -> 0 tan-1x/x  =  1

(6)  lim x -> a sin (x - a)/(x - a)  =  1

(7)  lim x -> a tan (x - a)/(x - a)  =  1

How to Evaluate Limits With Trig Functions - Examples

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 x

=  (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/) (1)

α/

Hence the value of lim  x -> 0 sin αx / sin x 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. After having gone through the stuff given above, we hope that the students would have understood, "How to Evaluate Limits With Trig Functions"

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