TS Inter 1st Year Maths 1B Limits and Continuity Important Questions Short Answer Type

Students must practice these Maths 1B Important Questions TS Inter 1st Year Maths 1B Limits and Continuity Important Questions Short Answer Type to help strengthen their preparations for exams.

TS Inter 1st Year Maths 1B Limits and Continuity Important Questions Short Answer Type

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. [May ’15 (AP), ’11]
Solution:
We find the limit at a = 1
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Q1
∴ f is continuous at x = 1
Hence f is continuous on R.

Question 2.
Is f defined by f(x) = \(\begin{cases}\frac{\sin 2 x}{x} & \text { if } x \neq 0 \\ 1, & \text { if } x=0\end{cases}\), continuous on ‘0’? [May ’12, ’10, ’04; Mar. ’05]
Solution:
Given, f(x) = \(\begin{cases}\frac{\sin 2 x}{x} & \text { if } x \neq 0 \\ 1, & \text { if } x=0\end{cases}\)
Take a = 0
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Q2
∴ f is discontinuous at x = 0.

TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type

Question 3.
Check the continuity of the following function at 2.
f(x) = \(\begin{cases}\frac{1}{2}\left(x^2-4\right) & \text { if } 0<x<2 \\ 0, & \text { if } x=2 \\ 2-8 x^{-3}, & \text { if } x>2\end{cases}\). [Mar. ’19 (TS): Mar. ’17 (AP): May ’15 (TS), ’08]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Q3
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Q3.1

Question 4.
Check the continuity of f given by f(x) = \(f(x)= \begin{cases}\frac{x^2-9}{x^2-2 x-3} & \text { if } 0<x<5 \text { and } x \neq 3 \\ 1.5 & \text { if } x=3\end{cases}\) at the point 3. [Mar. ’15 (AP), ’14, ’13, ’02; May ’04]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Q4

Question 5.
Prove that the functions sin x and cos x are continuous on R. [May ’08]
Solution:
(i) Let f(x) = sin x and a ∈ R
\(\lim _{x \rightarrow a} f(x)=\lim _{x \rightarrow a} \sin x\) = sin a = f(a)
∴ \(\lim _{x \rightarrow a} f(x)\) = f(a)
∴ f is continuous at x = a
∴ Since a is arbitrary, f is continuous on R.
(ii) Let g(x) = cos x and a ∈ R
\(\lim _{x \rightarrow a} g(x)=\lim _{x \rightarrow a} \cos x\) = cos a = g(a)
∴ \(\lim _{x \rightarrow a} g(x)\) = g(a)
∴ g is continuous at x = a
∴ since a is arbitrary, g is continuous on R.

Question 6.
Find real constants a, b so that the function f is given by f(x) = \(\begin{cases}\sin x & \text { if } x \leq 0 \\ x^2+\mathbf{a} & \text { if } 0<x<1 \\ \mathbf{b x}+3 & \text { if } 1 \leq x \leq 3 \\ -3 & \text { if } x>3\end{cases}\) is continuous on R. [Mar. ’18 (AP & TS); May ’13]
Solution:
Given, f(x) = \(\begin{cases}\sin x & \text { if } x \leq 0 \\ x^2+\mathbf{a} & \text { if } 0<x<1 \\ \mathbf{b x}+3 & \text { if } 1 \leq x \leq 3 \\ -3 & \text { if } x>3\end{cases}\)
Since f is continuous on R.
f is continuous at 0, 3.
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Q6
Since f is continuous at x = 3 then
LHL = RHL
3b + 3 = -3
3b = -3 – 3
b = -2
∴ a = 0, b = -2

TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type

Question 7.
Show that f(x) = \(\left\{\begin{array}{cl}
\frac{\cos a x-\cos b x}{x^2} & \text { if } x \neq 0 \\
\frac{1}{2}\left(b^2-a^2\right) & \text { if } x=0
\end{array}\right.\) where a and b are real constants, is continuous at ‘0’. [Mar. ’17 (TS), ’13(old), ’09; May ’14; B.P.]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Q7
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Q7.1

Some More Maths 1B Limits and Continuity Important Questions Short Answer Type

Question 8.
Find \(\lim _{x \rightarrow 3} \frac{x^3-6 x^2+x}{x^2-9}\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q1

Question 9.
Find \(\lim _{x \rightarrow 3} \frac{x^3-3 x^2}{x^2-5 x+6}\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q2

Question 10.
Compute \(\lim _{x \rightarrow 3} \frac{x^4-81}{2 x^2-5 x-3}\)
Solution:
Given, \(\lim _{x \rightarrow 3} \frac{x^4-81}{2 x^2-5 x-3}\)
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q3

Question 11.
Compute \(\lim _{x \rightarrow 3} \frac{x^2-8 x+15}{x^2-9}\). [Mar. ’16 (AP & TS)]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q4

Question 12.
If f(x) = \(-\sqrt{25-x^2}\) then find \(\lim _{x \rightarrow 1} \frac{f(x)-f(1)}{x-1}\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q5
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q5.1

TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type

Question 13.
Compute \(\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}\), b ≠ 0, a ≠ b. [Mar. ’18 (TS)]
Solution:
Given, \(\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}\)
Now dividing the numerator and denominator by x, we get
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q6

Question 14.
Compute \(\lim _{x \rightarrow 0} \frac{e^{3 x}-1}{x}\). [Mar. ’18 (AP); May ’15 (TS)]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q7

Question 15.
Evaluate \(\lim _{x \rightarrow 1} \frac{\log _e x}{x-1}\).
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q8

Question 16.
Compute \(\lim _{x \rightarrow 3} \frac{e^x-e^3}{x-3}\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q9

Question 17.
Compute \(\lim _{x \rightarrow 0} \frac{e^{\sin x}-1}{x}\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q10

TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type

Question 18.
Compute \(\lim _{x \rightarrow 1} \frac{(2 x-1)(\sqrt{x}-1)}{2 x^2+x-3}\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q11

Question 19.
Compute \(\lim _{x \rightarrow 0} \frac{\log _e(1+5 x)}{x}\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q12

Question 20.
Compute \(\lim _{x \rightarrow 0} \frac{(1+x)^{\frac{1}{8}}-(1-x)^{\frac{1}{8}}}{x}\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q13
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q13.1

Question 21.
Compute \(\lim _{x \rightarrow 0} \frac{1-\cos x}{x}\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q14

Question 22.
Compute \(\lim _{x \rightarrow 0} \frac{\sec x-1}{x^2}\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q15
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q15.1

TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type

Question 23.
Compute \(\lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}\), n ≠ 0.
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q16

Question 24.
Compute \(\lim _{x \rightarrow 0} \frac{x\left(e^x-1\right)}{1-\cos x}\). [May ’14]
Solution:
Given, \(\lim _{x \rightarrow 0} \frac{x\left(e^x-1\right)}{1-\cos x}\)
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q17

Question 25.
Compute \(\lim _{x \rightarrow 0} \frac{\log \left(1+x^3\right)}{\sin ^3 x}\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q18

Question 26.
Compute \(\lim _{x \rightarrow 0} \frac{x \tan 2 x-2 x \tan x}{(1-\cos 2 x)^2}\)
Solution:
Given, \(\lim _{x \rightarrow 0} \frac{x \tan 2 x-2 x \tan x}{(1-\cos 2 x)^2}\)
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q19

Question 27.
Compute \(\lim _{x \rightarrow \infty} \frac{x^2-\sin x}{x^2-2}\). [May ’16 (AP)]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q20
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q20.1

Question 28.
Compute \(\lim _{x \rightarrow 3} \frac{x^2+3 x+2}{x^2-6 x+9}\). [Mar. ’19 (AP)]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q21

TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type

Question 29.
Compute \(\lim _{x \rightarrow \infty} \frac{3 x^2+4 x+5}{2 x^3+3 x-7}\). [May ’15 (AP)]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q22

Question 30.
Compute \(\lim _{x \rightarrow \infty} \frac{6 x^2-x+7}{x+3}\).
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q23

Question 31.
Compute \(\lim _{x \rightarrow \infty} \frac{x^2+5 x+2}{2 x^2-5 x+1}\). [May ’14; Mar. ’17 (AP)]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q24

Question 32.
Compute \(\lim _{x \rightarrow 2}\left[\frac{1}{x-2}-\frac{4}{x^2-4}\right]\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q25
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q25.1

Question 33.
Compute \(\lim _{x \rightarrow-\infty} \frac{5 x^3+4}{\sqrt{2 x^4+1}}\)
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q26

TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type

Question 34.
Compute \(\lim _{x \rightarrow \infty} \frac{2+\cos ^2 x}{x+2007}\)
Solution:
Given, \(\lim _{x \rightarrow \infty} \frac{2+\cos ^2 x}{x+2007}\)
We know that
-1 ≤ cos x ≤ 1
0 ≤ cos2x ≤ 1
2 + 0 ≤ 2 + cos2x ≤ 2 + 1
2 ≤ 2 + cos2x ≤ 3
\(\frac{2}{x+2007} \leq \frac{2+\cos ^2 x}{x+2007} \leq \frac{3}{x+2007}\)
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q27

Question 35.
Compute \(\lim _{x \rightarrow-\infty} \frac{6 x^2-\cos 3 x}{x^2+5}\)
Solution:
We know that
-1 ≤ cos x ≤ 1
-1 ≥ cos 3x ≥ 1
1 ≥ -cos 3x ≥ -1
-1 ≤ -cos 3x ≤ 1
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q28

Question 36.
Show that f, given by f(x) = \(\frac{\mathbf{x}-|\mathbf{x}|}{\mathbf{x}}\) (x ≠ 0), is continuous on R – {0}.
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q29
∴ f is discontinuous at x = 0.
∴ Hence f is continuous on R – {0}.

Question 37.
If f is a function defined by f(x) = \(\begin{cases}\frac{x-1}{\sqrt{x}-1} & \text { if } x>1 \\ 5-3 x & \text { if }-2 \leq x \leq 1 \\ \frac{6}{x-10} & \text { if } x<-2\end{cases}\) then discuss the continuity of ‘f’.
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q30
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q30.1

TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type

Question 38.
If f, given by f(x) = \(\begin{cases}\mathbf{k}^2 x-k & \text { if } \mathbf{k} \geq 1 \\ 2 & \text { if } x<1\end{cases}\) is a continuous function on R, then find the values of k. [Mar. ’15 (TS)]
Solution:
Given, \(\begin{cases}\mathbf{k}^2 x-k & \text { if } \mathbf{k} \geq 1 \\ 2 & \text { if } x<1\end{cases}\)
∴ f is continuous on R
∴ f is continuous at x = 1
at x = 1, LHL = RHL = f(1) ………(1)
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q31
From (1), LHL = RHL
⇒ 2 = k2 – k
⇒ k2 – k – 2 = 0
⇒ k2 – 2k + k – 2 = 0
⇒ k(k – 2) + 1(k – 2) = 0
⇒ (k – 2)(k + 1) = 0
⇒ k – 2 = 0 (or) k + 1 = 0
⇒ k = 2 (or) k = -1

Question 39.
Check the continuity of ‘f’ given by f(x) = \(\left\{\begin{array}{rlr}
4-x^2, & \text { if } & x \leq 0 \\
\mathbf{x}-5, & \text { if } & 0 4 x^2-9, & \text { if } & 1<x<2 \\
3 x+4, & \text { if } & x \geq 2
\end{array}\right.\) at points x = 0, 1, 2. [Mar. ’16 (TS)]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q32
TS Inter First Year Maths 1B Limits and Continuity Important Questions Short Answer Type Some More Q32.1

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