**Estimating Limit Values From Graphs :**

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

lim x->x_{0} f(x) = L exists if the following hold :

(i) lim x->x_{0}+ f(x) exists,

(ii) lim x->x_{0}- f(x) exists, and

(iii) lim x->x_{0}+ f(x) = lim x->x_{0}- f(x) = L

**Question 1 :**

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

lim _{x->1} sin πx

**Solution :**

When 1 approaches from left hand side, we get the value closer to 0.

When 1 approaches from right hand side, we get the value closer to 0.

Hence the required limit 0.

**Question 2 :**

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

lim _{x->1} sec x

**Solution :**

When 1 approaches from left hand side, we get the value closer to 1.

When 1 approaches from right hand side, we get the value closer to 1.

Hence the required limit 1.

**Question 3 :**

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

lim _{x->}_{π/2} tan x

**Solution :**

When 1 approaches from left hand side, we get the value closer to 1.

When 1 approaches from right hand side, we get the value closer to 1.

Hence the function does not exist.

**Question 4 :**

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) = x^{2} + 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) = **(x^{2} + 2)**. It is enough to check if we get equal values for left hand and right hand limit.**

f(x) = (x lim = 3 |
f(x) = (x lim = 3 |

lim _{x->1}^{-} f(x) = lim _{x->1}+ f(x)

Hence the required limit is 3.

**Question 5 :**

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 6 :**

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, "Estimating Limit Values From Graphs"

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