HOW TO FIND LIMITS BY LOOKING AT A GRAPH

About "How to Find Limits by Looking at a Graph"

How to Find Limits by Looking at a Graph :

Here we are going to see how to find limits by looking at a graph.

Before going to see example problems, first let us know about left hand limit and right hand limit.

Left hand limit

We say that the left-hand limit of f(x) as x approaches x0 (or the limit of f(x) as x approaches from the left) is equal to l1 if we can make the values of f(x) arbitrarily close to l1 by taking x to be sufficiently close to x0 and less than x0. It is symbolically written as

f(x0-)  =  lim x ->x0f(x)  =  l1 Right hand limit

We say that the right-hand limit of f(x) as x approaches x0 (or the limit of f(x) as x approaches from the right) is equal to l2 if we can make the values of f(x) arbitrarily close to l2 by taking x to be sufficiently close to x0 and greater than x0. It is symbolically written as

f(x0+)  =  lim x ->x0f(x)  =  l2 From the above discussions we conclude that

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.

The picture given below will illustrate the concept.  There is no function to the left of x0. Hence the function is not defined.

Let us look into some examples based on the above concept.

Question 1 :

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

lim x->3 (4 - x) Solution :

 f(x)  =  (4 - x)lim x->3- f(x)  =  4 - 3  =  1 f(x)  =  (4 - x)lim x->3+ f(x)  =  4 - 3  =  1

lim x->3 f(x)  =  4 - 3  =  1

lim x->3- f(x)  =  lim x->3+ f(x)  =  lim x->3 f(x)

The function is defined at x -> 3. Hence the required limit is 1.

Question 2 :

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

lim x->1 (x2 + 2) Solution :

 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)  =  x2 + 2  =  12 + 2  =  3

lim x->1- f(x)  =  lim x->1+ f(x)  =  lim x->1 f(x)

The function is defined at x -> 1. Hence the required limit is 3 After having gone through the stuff given above, we hope that the students would have understood, "How to Find Limits by Looking at a Graph"

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