EVALUATING LIMITS SPECIAL TRIGONOMETRIC FUNCTIONS

There are three special trigonometric limits rules that we must know. We will learn more about these techniques in calculus.

limx→0sinxx=1limx→0cosx - 1x=0limx→0tanxx=1

Evaluate the following limits. Show all work.

Problem 1 :

x ⇾ 0 -2 sin x x

Solution :

Given, x ⇾ 0 -2 sin x x= -2 x ⇾ 0 sin x x= -2 (1)= -2

Problem 2 :

x ⇾ 0 sin x 3x

Solution :

Given, x ⇾ 0 sin x 3x= 13 x ⇾ 0 sin x x= 13 (1)= 13

Problem 3 :

x ⇾ 0 sin 5x x

Solution :

Given, x ⇾ 0 sin 5x xMultiplying and dividing by 5.= 5 x ⇾ 0 sin 5x 5x= 5 5x ⇾ 0 sin 5x 5x= 5 (1)= 5

Problem 4 :

x ⇾ 0 3 cos x - 3 x

Solution :

Given, x ⇾ 0 3 cos x - 3 x= x ⇾ 0 3(cos x - 1) x= 3 x ⇾ 0 cos x - 1 x= 3 (0)= 0

Problem 5 :

x ⇾ 0 3 tan x 2x

Solution :

Problem 6 :

x ⇾ 0 cos x - 1 3x

Solution :

Given, x ⇾ 0 cos x - 1 3x= 13x ⇾ 0 cos x - 1 x= 13 (0)= 0

Problem 7 :

x ⇾ 0 sin 3x 2x

Solution :

Given, x ⇾ 0 sin 3x 2xMultiplying and dividing by 33.= x ⇾ 0 33 ⋅sin 3x 2x= 32 x ⇾ 0 sin 3x 3x= 32 3x ⇾ 0 sin 3x 3x= 32 (1)= 32

Problem 8 :

x ⇾ 0 x4 sin x

Solution :

Given, x ⇾ 0 x4 sin x= 14 x ⇾ 0 x sin x= 14 (1)= 14

Problem 9 :

x ⇾ 0 sin 6x x

Solution :

Given, x ⇾ 0 sin 6x xMultiplying and dividing by 6.= 6 x ⇾ 0 sin 6x 6x= 6 6x ⇾ 0 sin 6x 6x= 6 (1)= 6

Problem 10 :

x ⇾ 0- sin xx + 1x

Solution :

Given, x ⇾ 0- sin xx + 1x= x ⇾ 0- sin xx + x ⇾ 0- 1x= 1 + (-∞)= -∞

Problem 11 :

x ⇾ 0 x sin 7x x

Solution :

Given, x ⇾ 0 x sin 7x xMultiplying and dividing by 7.= 7 x ⇾ 0 x sin 7x 7x= 7 (0)= 0

Problem 12 :

x ⇾ 0 3x + sin 3xx

Solution :

Given,x ⇾ 0 3x + sin 3xx= x ⇾ 0 3x + x ⇾ 0 sin 3xxx ⇾ 0 sin 3xxMultiplying and dividing by 3.=3 x ⇾ 0 sin 3x3x= 3 3x ⇾ 0 sin 3x3x= 3(1) = 3= 0 + 3= 3

Recent Articles

RSS

  1. Finding Range of Values Inequality Problems

    May 21, 24 08:51 PM

    Finding Range of Values Inequality Problems

    Read More

  2. Solving Two Step Inequality Word Problems

    May 21, 24 08:51 AM

    Solving Two Step Inequality Word Problems

    Read More

  3. Exponential Function Context and Data Modeling

    May 20, 24 10:45 PM

    Exponential Function Context and Data Modeling

    Read More