HOW TO ESTIMATE LIMITS FROM GRAPHS

About "How to Estimate Limits From Graphs"

How to Estimate Limits From Graphs :

Here we are going to see how to estimate limits from graph.

Required Condition for Existence of Limit of Function

lim x->x0 f(x)  =  L exists if the following hold :

(i) lim x->x0+ f(x) exists, 

(ii)  lim x->x0- f(x) exists, and

(iii)  lim x->x0+ f(x)  =   lim x->x0- f(x)  =  L

How can we say the function is not exists by looking at its graph ?

When we get different values as x0 approaches from left and from right, we may say that the function does not exists.

Question 1 :

Use the graph to find the limits (if it exists). If the limit does not exist, explain why?

lim x->2 f(x)  

Where f(x)  =  4 - x   ≠ 2

                       0          x = 2

Solution :

To find the value of left hand limit and right hand limit for x -> 2, we have to use the function f(x)  =  4 - x. It is enough to check if we get equal values for left hand and right hand limit.

f(x)  =  4 - x

lim x->2- f(x)  =  4 - 2

  =  2

f(x)  =  (4 - x)

lim x->2+ f(x)  =  4 - 2

  =  2

f(x)  =  0 at x = 2

lim x->2- f(x)  =  lim x->2+ f(x) 

Hence the required limit 2.

Question 2 :

Use the graph to find the limits (if it exists). If the limit does not exist, explain why?

lim x->1 f(x) 

Where f(x)  =  x2 + 2      x ≠ 1

                    =    1            x = 1

Solution :

To find the value of left hand limit and right hand limit for x -> 1, we have to use the function f(x)  =  (x2 + 2). It is enough to check if we get equal values for left hand and right hand limit.

f(x)  =  (x2 + 2)

lim x->1- f(x)  =  12 + 2

  =  3

f(x)  =  (x2 + 2)

lim x->1+ f(x)  =  12 + 2

  =  3

lim x->1- f(x)  =  lim x->1+ f(x)

Hence the required limit is 3.

Question 3 :

Use the graph to find the limits (if it exists). If the limit does not exist, explain why?

lim x->3 1/(x- 3)

Solution :

From the graph given above, we get different values for left hand limit and right hand limit.

The function does not exist at x - >3.

Question 4 :

Use the graph to find the limits (if it exists). If the limit does not exist, explain why?

lim x->5 |x - 5|/(x - 5)

Solution :

From the graph given above, we get different values for left hand limit and right hand limit.

The function does not exist at x - >5.

After having gone through the stuff given above, we hope that the students would have understood, "How to Estimate Limits From Graphs"

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