Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e)

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:
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 1
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:
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 2
f is not continuous at 0

Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e)

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.
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 3
Solution:
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 4
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:
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 5
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
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 6
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 7

If x = 0, f(a) is not defined
f(x) is not continuous at ’0′
∴ f(x) is continuous on R – {0}

Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e)

Question 4.
If f is a function defined by
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 8
then discuss the continuity of f.
Solution:
Case (i) : x = 1
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 9
f(x) is not continuous at x > 1

Case (ii) : x = -2
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 10
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:
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 11
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
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 12
∴ f is continuous at a.

ii) Let a ∈ R
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 13
∴ f is continuous at a.

III.

Question 1.
Check the continuity of ‘f given by
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 14
at the points 0, 1 and 2.
Solution:
i) Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 15
∴ f(x) is continuous at x = 0

ii) Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 16
∴ f(x) is continuous at x = 1

iii) Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 17
∴ f(x) is continuous at x = 2

Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e)

Question 2.
Find real constant a, b so that the function f given by
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 18
is continuous on R.
Solution:
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 19
Since f(x) is continuous on R
LHS = RHS ⇒ a = 0
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 20
Since f(x) is continuous on R.
LHS = RHS
3b + 3 = -3
3b = – 6 ⇒ b = -2

Question 3.
Show that
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 21
where a and b are real constants, is continuous at 0.
Solution:
Inter 1st Year Maths 1B Limits and Continuity Solutions Ex 8(e) 22
∴ f(x) is continuous at x = 0