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Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5

Question 1.
Prove that f(x) = 2x2 + 3x – 5 is continuous at all points in R.
Solution:
Polynomial functions are continuous at every points of R.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 1

Question 2.
Examine the continuity of the following:
(i) x + sin x
Solution:
f(x) = x + sin x
The Domain of the function (-∞, ∞)
∴ f(x) is continuous in (-∞, ∞)(i.e.,) for all x ∈ R

(ii) x2 cos x
Solution:
f(x) = x2 cos x
The Domain of the function (-∞, ∞)
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 2
f(x) is continuous in R

(iii) ex tan x
Solution:
The Domain of the function in R – {(2n + 1) π/2}
∴ The functions is continuous for all x ∈ R – (2n + 1) \(\frac{\pi}{2}\), n ∈ Z

(iv) e2x + x2
f(x) = e2x + x2 = 1 + 2x + \(\frac{(2 x)^{2}}{2 !}\) + …………. + x2
Solution:
∴ The functions is continuous for all x ∈ R

(v) x.ln x
Solution:
Thus f(x) is continuous for (0, ∞)

(vi) \(\frac{\sin x}{x^{2}}\)
Solution:
Thus f(x) is continuous for all x ∈ R – {0}

(vii) \(\frac{x^{2}-16}{x+4}\)
Solution:
f(x) = \(\frac{x^{2}-16}{x+4}=\frac{(x-4)(x+4)}{x+4}\)
The function f(x) is continuous for all x ∈ R – {-4}

(viii) |x + 2| + |x – 1|
Solution:
f(x) is continuous for x ∈ R

(ix) \(\frac{|x-2|}{|x+1|}\)
Solution:
The function is continuous for all x ∈ R – {-1}

(x) cot x + tan x
Solution:
The function is continuous for all x ∈ R – \(\frac{n \pi}{2}\), n ∈ z.

Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5

Question 3.
Find the points of discontinuity of the function f, where,
(i)
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 3
Solution:
f(3) = 12 + 5 = 17
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 4
∴ f(x) is discontinuous at x = 3

(ii)
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 5
Solution:
f(x) = 4
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 6
∴ f(x) is continuous for all x ∈ R

(iii)
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 7
Solution:
f(x) = 8 – 3 = 5
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 8
∴ f(x) is continuous for all x ∈ R

(iv)
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 9
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 10
∴ f(x) is continuous for all x ∈ [0, π/2]

Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5

Question 4.
At the given points x0 discover whether the given function is continuous or discontinuous citing the reasons for your answer.
(i)
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 11
Solution:
Given f(x0) = 1
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 12
∴ f(x) is continuous at x0 = 1

(ii)
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 13
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 14
∴ f(x) is not continuous at x0 = 3

Question 5.
Show that the function Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 15 is continuous on (-∞, ∞)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 16
Given that f(1) = 3
∴ f(x) is continuous for all x ∈ R

Question 6.
For what value of α is this function f(x) = Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 17 continuous at x = 1?
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 18
∵ f(x) is continuous at x = 1, α = 4

Question 7.
Let Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 19 Graph the function. Show that f(x) continuous on (-∞, ∞)
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 20
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 21
∴ f(x) is continuous in (-∞, ∞)

Question 8.
If f and g are continuous function with f(3) = 5 and Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 22 find g(3).
Solution:
Since f and g are continuous
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 23
2f(3) – g(3) = 4
2(5) – g(3) = 4
10 – 4 = g(3)
g(3) = 6

Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5

Question 9.
Find the points at which f is discontinuous. At which of these points f is continuous from the right, from the left, or neither? Sketch the graph of f.
(i)
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 24
∴ f(x) is not continuous at x = 1
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 25
f(x) is not continuous at x = 1

(ii)
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 26
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 27
∴ f(x) is not continuous at x = 0

Question 10.
A function f is defined as follows:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 28
Is the function continuous?
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 29
From (i), (ii) and (iii)
f(x) is continuous at x = 0, 1, 3

Question 11.
Which of the following functions f has removable discontinuity at x = x0? If the discontinuity is removable, find a function g that agrees with f for x ≠ x0 and is continuous on R.
(i) f(x) = \(\frac{x^{2}-2 x-8}{x+2}\), x0 = -2
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 30

(ii) f(x) = \(\frac{x^{3}+64}{x+4}\), x0 = -4
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 31
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 32

(iii) f(x) = \(\frac{3-\sqrt{x}}{9-x}\), x0 = 9
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 33

Question 12.
Find the constant b that makes g continuous on (-∞, ∞)
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 34
Solution:
Since g(x) is continuous,
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 35

Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5

Question 13.
Consider the function f(x) = x sin \(\frac{\pi}{x}\). What value must we give f(0) in order to make the function continuous everywhere?
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 36
so to make the function f(x) is continuous at f(0) = 0

Question 14.
The function f(x) = \(\frac{x^{2}-1}{x^{3}-1}\) is not defined at x = 1. What value must we give f(1) in order to make f(x) continuous at x = 1?
Solution:
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 37

Question 15.
State how continuity is destroyed at x = x0 for each of the following graphs.
Samacheer Kalvi 11th Maths Solutions Chapter 9 Limits and Continuity Ex 9.5 38
Solution:

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