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

Question 1.

Prove that f(x) = 2x^{2} + 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) x^{2} cos x

Solution:

f(x) = x^{2} cos x

The Domain of the function (-∞, ∞)

f(x) is continuous in R

(iii) e^{x} 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) e^{2x} + x^{2}

f(x) = e^{2x} + x^{2} = 1 + 2x + \(\frac{(2 x)^{2}}{2 !}\) + …………. + x^{2}

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 x_{0} discover whether the given function is continuous or discontinuous citing the reasons for your answer.

(i)

Solution:

Given f(x_{0}) = 1

∴ f(x) is continuous at x_{0} = 1

(ii)

Solution:

∴ f(x) is not continuous at x_{0} = 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 a removable discontinuity at x = x_{0}? If the discontinuity is removable, find a function g that agrees with f for x ≠ x_{0} and is continuous on R.

(i) f(x) = \(\frac{x^{2}-2 x-8}{x+2}\), x_{0} = -2

Solution:

(ii) f(x) = \(\frac{x^{3}+64}{x+4}\), x_{0} = -4

Solution:

(iii) f(x) = \(\frac{3-\sqrt{x}}{9-x}\), x_{0} = 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 = x_{0} for each of the following graphs.

Solution: