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

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

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

Question 1.
Find \(\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1}{x}\). [Mar. ’14, ’07, ’04; May ’10]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q1

Question 2.
Compute \(\lim _{x \rightarrow 0}\left[\frac{e^x-1}{\sqrt{1+x}-1}\right]\). [Mar. ’15 (TS), ’13 (0ld), ’09]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q2

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

Question 3.
Compute \(\lim _{x \rightarrow 0} \frac{\mathbf{a}^x-1}{\mathbf{b}^x-1}\), (a > 0), (b > 0), (b ≠ 1). [Mar. ’19 (TS); Mar. ’15 (AP), ’13, ’08, ’02; May ’02]
Solution:
Given \(\lim _{x \rightarrow 0} \frac{\mathbf{a}^x-1}{\mathbf{b}^x-1}\)
Now dividing Numerator and Denominator by x, we get
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q3

Question 4.
Compute \(\lim _{x \rightarrow 0} \frac{e^x-\sin x-1}{x}\). [Mar. ’16 (TS), ’13]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q4

Question 5.
Compute \(\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{\left(x-\frac{\pi}{2}\right)}\). [Mar. ’18 (TS); May ’13 (old), ’05; Mar. ’08]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q5

Question 6.
Compute \(\lim _{x \rightarrow 0} \frac{\sin a x}{x \cos x}\). [Mar. ’03, ’02]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q6

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

Question 7.
Compute \(\lim _{x \rightarrow 1} \frac{\sin (x-1)}{x^2-1}\). [May ’06, ’02]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q7

Question 8.
Compute \(\lim _{x \rightarrow 0} \frac{\sin (a+b x)-\sin (a-b x)}{x}\). [Mar. ’12, ’08, ’05; May ’09]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q8
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q8.1

Question 9.
Compute \(\lim _{x \rightarrow a} \frac{\tan (x-a)}{x^2-a^2}\) (a ≠ 0) [May ’04, Mar. ’15 (TS)]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q9

Question 10.
Compute \(\lim _{x \rightarrow 0} \frac{e^{7 x}-1}{x}\). [Mar. ’17 (AP); May ’13]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q10

Question 11.
Compute \(\lim _{x \rightarrow 0} \frac{e^{3+x}-e^3}{x}\). [Mar. ’19 (AP); B.P.]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q11

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

Question 12.
Compute \(\lim _{x \rightarrow a} \frac{x \sin a-a \sin x}{x-a}\). [Mar. ’11; Mar. ’19 (AP); Mar. ’16 (AP)]
Solution:
Given \(\lim _{x \rightarrow a} \frac{x \sin a-a \sin x}{x-a}\)
Adding and subtracting, a sin a, in the numerator, We get
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q12
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q12.1

Question 13.
Compute \(\lim _{x \rightarrow 0} \frac{\cos a x-\cos b x}{x^2}\). [May ’11; Mar. ’07, ’04]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q13
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q13.1

Question 14.
Compute \(\lim _{x \rightarrow 2} \frac{2 x^2-7 x-4}{(2 x-1)(\sqrt{x}-2)}\). [May ’12, ’07]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q14

Question 15.
Compute \(\lim _{x \rightarrow 0} \frac{3^x-1}{\sqrt{1+x}-1}\). [Mar. ’05]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q15

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

Question 16.
Compute \(\lim _{x \rightarrow 0} \frac{\sqrt[3]{1+x}-\sqrt[3]{1-x}}{x}\). [May ’15 (AP), ’06, ’97, ’02; Mar. ’93]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q16

Question 17.
Compute \(\lim _{x \rightarrow a} \frac{\sin (x-a) \tan ^2(x-a)}{\left(x^2-a^2\right)^2}\). [Mar. ’06]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q17
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q17.1

Question 18.
Compute \(\lim _{x \rightarrow 0} \frac{1-\cos 2 m x}{\sin ^2 n x}\). [Mar. ’10; May ’15 (TS)]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q18

Question 19.
Show that \(\lim _{\mathbf{x} \rightarrow 0^{-1}} \frac{|\mathbf{x}|}{\mathbf{x}}=1\) and \(\lim _{\mathbf{x} \rightarrow 0^{+1}} \frac{|\mathbf{x}|}{\mathbf{x}}=-1\). [Mar. ’93, ’85; May ’86, ’83]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q19

Question 20.
Show that \(\lim _{x \rightarrow 2^{-}} \frac{|x-2|}{x-2}=-1\). [May ’04]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q20

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

Question 21.
Show that \(\lim _{x \rightarrow 0^{+}} \frac{2|x|}{x}\) + x + 1 = 3. [Mar. ’15 (AP); May ’08; B.P.]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q21

Question 22.
Compute \(\lim _{x \rightarrow \infty} \frac{8|x|+3 x}{3|x|-2 x}\). [Mar. ’17 (TS), ’12; May ’10, ’09]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q22

Question 23.
Compute \(\lim _{x \rightarrow \infty} \frac{11 x^3-3 x+4}{13 x^3-5 x^2-7}\). [Mar. ’18 (AP); Mar. ’14; May ’07]
Solution:
Given \(\lim _{x \rightarrow \infty} \frac{11 x^3-3 x+4}{13 x^3-5 x^2-7}\)
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q23

Question 24.
Compute \(\lim _{x \rightarrow \infty}(\sqrt{x+1}-\sqrt{x})\). [May ’13]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q24

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

Question 25.
Compute \(\lim _{x \rightarrow \infty}\left(\sqrt{x^2+x}-x\right)\). [Mar. ’11, ’10, ’09, ’08]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q25
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q25.1

Question 26.
Compute \(\lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}\right)\). [Mar. ’17 (TS)]
Solution:
TS Inter First Year Maths 1B Limits and Continuity Important Questions Very Short Answer Type Q26

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