HOW TO FIND LEFT AND RIGHT LIMITS

About "How to Find Left and Right Limits"

How to Find Left and Right Limits :

Here we are going to see how to find left and right limits.

Question 1 :

(1) (a) Find the left and right limits of

f(x)  =  (x2 - 4) / (x2 + 4x + 4) (x + 3) at x  =  -2 

Solution :

f(x)  =  (x2 - 4) / (x2 + 4x + 4) (x + 3)

f(x)  =  (x + 2)(x - 2) / (x + 2)(x + 2)(x  + 3)

f(x)  =  (x - 2)/(x + 2)(x  + 3)

= lim x->-2-(x-2)/(x+2)(x +3)

To find left hand limit of -2-, let us take -2.1

  = (-2.1-2)/(-2.1+2)(-2.1+3)

  =  -4.1/(-0.1)(0.9)

  =  

To find left hand limit of -2+, let us take -1.9

  = (-1.9-2)/(-1.9+2)(-1.9+3)

  =  -3.9/(0.1)(1.1)

  =  -

Hence the left hand limit of -2- is ∞ and right hand limit of -2+ is -∞.

(b)  f (x) = tan x at x = π/2

Solution :

f (x) = tan x

f(x)  =  lim x->π/2tan x

The angle lesser than 90 degree lies in the first quadrant, for all trigonometric ratios we will get positive sign.

f(x)  =  lim x->π/2tan (π/2)

=  

The angle greater than 90 degree lies in the second quadrant, for sin and cosec, we will have positive and for tan we will get negative sign.

f(x)  =  lim x->π/2tan (π/2)

=  -

Question 2 :

Evaluate the following

 lim x->3 (x2 - 9) / x2(x2 - 6x + 9)

Solution :

 =  lim x->3 (x2 - 9) / x2(x2 - 6x + 9)

 =  lim x->3 (x + 3)(x - 3) / x2(x-3)2

 =  lim x->3 (x + 3) / x2(x-3)

The simplified form does not match with any formulas in limits, so let us find left hand and right hand limit.

Left hand limit :

  =  lim x->3- (x+3)/x2(x-3)

  =  -

Right hand limit :

=  lim x->3+ (x+3)/x2(x-3)

  = 

Question 3 :

lim x-> [3/(x - 2) - (2x + 11)/(x2 + x - 6)]

Solution :

f(x)  =  3/(x - 2) - (2x + 11)/(x2 + x - 6)

f(x)  =  [3/(x - 2) - (2x + 11)/(x-2) (x+3)]

=  3(x+3)-(2x+11)/(x-2) (x+3)

=  (3x+9-2x-11)/(x-2) (x+3)

=  (x - 2)/(x - 2)(x + 3)

=  1/(x + 3)

=  lim x->[1/(x + 3)]

Dividing by the highest exponents of denominator, we get

=  lim x->[(1/x)/(1 + 3/x)]

By applying ∞ instead of x, we get 

=  0

After having gone through the stuff given above, we hope that the students would have understood, "How to Find Left and Right Limits"

Apart from the stuff given in "How to Find Left and Right Limits" if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. SAT Math Videos

    May 22, 24 06:32 AM

    sattriangle1.png
    SAT Math Videos (Part 1 - No Calculator)

    Read More

  2. Simplifying Algebraic Expressions with Fractional Coefficients

    May 17, 24 08:12 AM

    Simplifying Algebraic Expressions with Fractional Coefficients

    Read More

  3. The Mean Value Theorem Worksheet

    May 14, 24 08:53 AM

    tutoring.png
    The Mean Value Theorem Worksheet

    Read More