Practicing the Intermediate 1st Year Maths 1B Textbook Solutions Inter 1st Year Maths 1B Limits and Continuity Solutions Exercise 8(e) will help students to clear their doubts quickly.

## Intermediate 1st Year Maths 1B Limits and Continuity Solutions Exercise 8(e)

I.

Question 1.

Is the function f, defined by \(f(x)=\left\{\begin{array}{l}

x^{2} \text { if } x \leq 1 \\

x \text { if } x>1

\end{array}\right.\) continuous on R?

Solution:

f is continuous at x = 1

f is continuous on R.

Question 2.

Is f defined by f(x) = \(=\left\{\begin{array}{cc}

\frac{\sin 2 x}{x}, & \text { if } x \neq 0 \\

1 & \text { if } x=0

\end{array}\right.\) continuous at 0?

Solution:

f is not continuous at 0

Question 3.

Show that the function f(x) = [cos (x^{10} + 1)]^{1/3}, x ∈ R is a continuous function.

Solution:

We know that cos x is continuous for every x ∈ R

∴ The given function f(x) is continuous for every x ∈ R.

II.

Question 1.

Check the continuity of the following function at 2.

Solution:

f(x) is not continuous at 2.

Question 2.

Check the continuity of f given by f(x) = \(\begin{cases}\frac{\left[x^{2}-9\right]}{\left[x^{2}-2 x-3\right]} & \text { if } 0<x<5 \text { and } x \neq 3 \\ 1.5 & \text { if } x=3\end{cases}\) at the point 3.

Solution:

f(x) is continuous at x = 3.

Question 3.

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

Solution:

Case (i) : a > 0 ⇒ |a| = a

If x = 0, f(a) is not defined

f(x) is not continuous at ’0′

∴ f(x) is continuous on R – {0}

Question 4.

If f is a function defined by

then discuss the continuity of f.

Solution:

Case (i) : x = 1

f(x) is not continuous at x > 1

Case (ii) : x = -2

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

Question 5.

If f is given by f(x) = \(=\left\{\begin{array}{cl}

k^{2} x-k & \text { if } x \geq 1 \\

2 & \text { if } x<1

\end{array}\right.\) is a continuous function on R, then find the values of k.

Solution:

2 = k² – k

k² – k – 2 = 0

(k – 2) (k + 1) = 0

k = 2 or – 1

Question 6.

Prove that the functions ‘sin x’ and ‘cos x’ are continuous on R.

Solution:

i) Let a ∈ R

∴ f is continuous at a.

ii) Let a ∈ R

∴ f is continuous at a.

III.

Question 1.

Check the continuity of ‘f given by

at the points 0, 1 and 2.

Solution:

i)

∴ f(x) is continuous at x = 0

ii)

∴ f(x) is continuous at x = 1

iii)

∴ f(x) is continuous at x = 2

Question 2.

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

is continuous on R.

Solution:

Since f(x) is continuous on R

LHS = RHS ⇒ a = 0

Since f(x) is continuous on R.

LHS = RHS

3b + 3 = -3

3b = – 6 ⇒ b = -2

Question 3.

Show that

where a and b are real constants, is continuous at 0.

Solution:

∴ f(x) is continuous at x = 0