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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.
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 (-∞, ∞)
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.
Question 3.
Find the points of discontinuity of the function f, where,
(i)
Solution:
f(3) = 12 + 5 = 17
∴ f(x) is discontinuous at x = 3
(ii)
Solution:
f(x) = 4
∴ f(x) is continuous for all x ∈ R
(iii)
Solution:
f(x) = 8 – 3 = 5
∴ f(x) is continuous for all x ∈ R
(iv)
Solution:
∴ f(x) is continuous for all x ∈ [0, π/2]
Question 4.
At the given points x0 discover whether the given function is continuous or discontinuous citing the reasons for your answer.
(i)
Solution:
Given f(x0) = 1
∴ f(x) is continuous at x0 = 1
(ii)
Solution:
∴ f(x) is not continuous at x0 = 3
Question 5.
Show that the function is continuous on (-∞, ∞)
Solution:
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) = continuous at x = 1?
Solution:
∵ f(x) is continuous at x = 1, α = 4
Question 7.
Let Graph the function. Show that f(x) continuous on (-∞, ∞)
Solution:
∴ f(x) is continuous in (-∞, ∞)
Question 8.
If f and g are continuous function with f(3) = 5 and find g(3).
Solution:
Since f and g are continuous
2f(3) – g(3) = 4
2(5) – g(3) = 4
10 – 4 = g(3)
g(3) = 6
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)
∴ f(x) is not continuous at x = 1
Solution:
f(x) is not continuous at x = 1
(ii)
Solution:
∴ f(x) is not continuous at x = 0
Question 10.
A function f is defined as follows:
Is the function continuous?
Solution:
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:
(ii) f(x) = \(\frac{x^{3}+64}{x+4}\), x0 = -4
Solution:
(iii) f(x) = \(\frac{3-\sqrt{x}}{9-x}\), x0 = 9
Solution:
Question 12.
Find the constant b that makes g continuous on (-∞, ∞)
Solution:
Since g(x) is continuous,
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:
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:
Question 15.
State how continuity is destroyed at x = x0 for each of the following graphs.
Solution: