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Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1

In problems 1-6, complete the table using calculate and use the result to estimate the limit.
Question 1.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 1
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 2

Question 2.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 3
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 4
∴ Limit is 0.25

Question 3.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 5
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 6
∴ Limit is 0.288

Question 4.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 7
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 8
∴ Limit is -0.25

Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1

Question 5.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 9
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 10
∴ Limit is 1

Question 6.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 11
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 12
∴ Limit is 0

In exercise problems 7-15, use the graph to find the limits (if it exists). If the limit does not exist, explain why?
Question 7.
\(\lim _{x \rightarrow 3}\)(4 – x)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 13
Limit exist and is equal to 1

Question 8.
\(\lim _{x \rightarrow 1}\)(x2 + 2)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 14
Limit exist and is equal to = 3

Question 9.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 15
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 16

Question 10.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 17
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 18

Question 11.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 19
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 20
Limit does not exist

Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1

Question 12.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 21
Solution:
When x → 5, (x – 5) = -(x – 5)
∴ \(\lim _{x \rightarrow 5^{-}} \frac{-(x-5)}{(x-5)}\) = -1
When x → 5+, (x – 5) = (x – 5)
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 22
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 23

Question 13.
\(\lim _{x \rightarrow 1}\) sin(πx)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 24
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 25

Question 14.
\(\lim _{x \rightarrow 0}\) (sec x)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 26

Question 15.
\(\lim _{x \rightarrow \frac{\pi}{2}}\) tan x
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 27
Limit does not exist

Sketch the graph of f, then identify the values of x0 for which \(\lim _{x \rightarrow x_{0}}\) f(x) exists.
Question 16.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 28
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 29

Question 17.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 30
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 31
Limit exists except at x0 = π

Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1

Question 18.
Sketch the graph of a function f that satisfies the given values:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 32
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 33

Question 19.
Write a brief description of the meaning of the notation \(\lim _{x \rightarrow 8}\) f(x) = 25
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 34

Question 20.
If f(2) = 4, can you conclude anything about the limit of f(x) as x approaches 2?
Solution:
No, f(x) = 4, It is the value of the function at x = 2
This limit doesn’t exists at x = 2
Since f(2) = 4
It need not imply that \(\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)\)
∴ we cannot conclude at x = 2

Question 21.
If the limit of f(x) as z approaches 2 is 4, can you conclude anything about f(2)?
Explain reasoning.
Solution:
\(\lim _{x \rightarrow 2}\) f(x) 4 , \(\lim _{x \rightarrow 2^{-}}\) f(x) = \(\lim _{x \rightarrow 2^{+}}\) f(x) = 4
When x approaches 2 from the left or from the right f(x) approaches 4.
Given that \(\lim _{x \rightarrow 2^{-}}\) f(x) = \(\lim _{x \rightarrow 2^{+}}\) f(x) = 4
The existence or non existence at x =2 has no leaving on the existence of the limit of f(x) as x approaches to 2.
∴ We cannot conclude the value of f(2)

Question 22.
Evaluate: \(\lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3}\) if it exists by finding f(3) and f(3+).
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 35
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 36

Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1

Question 23.
Verify the existence of Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 37
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 38
Limit does not exist

Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 Additional Questions

Question 1.
Suppose Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 39. What are possible values of a and b?
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 40

Question 2.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 41
Solution:
We have,
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 42

Question 3.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 43
Solution:
We have,
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 44

Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1

Question 4.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 45
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 46
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 47

Question 5.
Let a1, a2 …………… an be fixed real numbers such that f(x) = (x – a1) , (x – a2), ………. (x – an) what \(\lim _{x \rightarrow a}\) f(x) For a ≠ a1, a2, ………… an compute \(\lim _{x \rightarrow a}\) f(x)
Solution:
We have,
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.1 48

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