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 (x10 + 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