How to Estimate Limits From Graphs :
Here we are going to see how to estimate limits from graph.
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
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 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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