Students must practice these TS Inter 1st Year Maths 1B Study Material Chapter 8 Limits and Continuity Ex 8(e) to find a better approach to solving the problems.

## TS Inter 1st Year Maths 1B Limits and Continuity 8(e)

Question 1.

Is the function f, defined by

continuous on R? (V.S.A.Q.) (May 2011)

Answer:

The function is continuous at ‘1’ hence at continuous on R.

Question 2.

Is f defined by

continuous at ‘0’ (V.S.A.Q.) (May 2012)

Answer:

Given f(0) = 1

Hence f is continuous at ‘0’.

Question 3.

Show that the function

f(x) = [cos(x^{10} + 1)]^{\(\frac{1}{3}\)}; x ∈ ℛ is a continuous function. (V.S.A.Q.)

Answer:

Since cos x is continuous for every x ∈ R, we have f(x) = [cos (x^{10} + 1)]^{1/3} is also continuous over R.

II.

Question 1.

Check the continuity of the following function at 2. (S.A.Q.)

Answer:

Given f(2) = 0

Hence f(x) is not continuous at x = 2.

Question 2.

Check the continuity of f given by (S.A.Q.) (Mar. ’14, ’13)

Answer:

Given f(3) = 1.5 and

Since \(\lim _{x \rightarrow 3}\) f(x) = f(3) we have f is continuous at the point ‘3’.

Question 3.

Show that f given by f(x) = \(\frac{\mathbf{x}-|\mathbf{x}|}{\mathbf{x}}\) (x ≠ 0) is continuous on R – {0}. (S.A.Q.)

Answer:

In both the cases \(\lim _{x \rightarrow a}\) f(x) = f(a)

But if a = 0 then f(0) is not defined f(x) is not continuous at ‘0’.

Hence f(x) is continuous on R – {0}

Question 4.

If f is a function defined by (S.A.Q.)

then discuss the continuity of f.

Answer:

Question 5.

If f is given by

is a continuous function on R, then find the values of ‘k’. (S.A.Q.)

Answer:

Given that f is continuous on R, then it is continuous at ‘1’.

∴ k^{2} – k = 2 ⇒ k^{2} – k – 2 = 0

⇒ (k – 2) (k + 1) = 0

⇒ k = 2 or k = – 1

Question 6.

Prove that the function ‘sin x’ and ‘cos x’ are continuous on R. (S.A.Q.)

Answer:

(i) Let f(x) = sin x

If f is continuous at a point ‘a’ then

\(\lim _{x \rightarrow \mathrm{a}}\) f(x) = f(a)

∴ \(\lim _{x \rightarrow \mathrm{a}}\) (sin x) = sin a = f(a)

(or) by definition we have for ∈ > 0, ∃ a δ > 0

such that |sin x – sin a| < E for |x – a| < δ

Hence f(x) = sin x is continuous at ‘a’

Hence f is continuous over R.

(ii) f(x) = cos x

Let a ∈ R,

∴ \(\lim _{x \rightarrow a}\) f(x) = \(\lim _{x \rightarrow a}\) (cos x) = cos a = f(a)

∴ f is continuous at ‘a’

Hence f is continuous over R.

III.

Question 1.

Check the continuity of f given by

at the point 0, 1 and 2.

Answer:

Question 2.

Find real constants a, b so that the function f given by

is continuous on R

Answer:

Given f is continuous over R ⇒ f is continuous at 0, 1 and 3.

If f is continuous at ‘0’ then

∴ In all cases if f is continuous at 0, 1, 3 and continuous over R.

The values are a = 0,

b = – 2.

Question 3.

Show that

where a and b are real constants, is continuous at ‘0’. (Board New Model Paper) (E.Q.)

Answer: