{"id":4235,"date":"2023-12-22T15:45:05","date_gmt":"2023-12-22T10:15:05","guid":{"rendered":"https:\/\/tsboardsolutions.in\/?p=4235"},"modified":"2023-12-23T17:30:44","modified_gmt":"2023-12-23T12:00:44","slug":"maths-1b-limits-and-continuity-important-questions-short-answer-type","status":"publish","type":"post","link":"https:\/\/tsboardsolutions.in\/maths-1b-limits-and-continuity-important-questions-short-answer-type\/","title":{"rendered":"TS Inter 1st Year Maths 1B Limits and Continuity Important Questions Short Answer Type"},"content":{"rendered":"

Students must practice these Maths 1B Important Questions<\/a> TS Inter 1st Year Maths 1B Limits and Continuity Important Questions Short Answer Type to help strengthen their preparations for exams.<\/p>\n

TS Inter 1st Year Maths 1B Limits and Continuity Important Questions Short Answer Type<\/h2>\n

Question 1.
\nIs the function f, defined by f(x) = \\(\\left\\{\\begin{array}{l}
\nx^2 \\text { if } x \\leq 1 \\\\
\nx \\text { if } x>1
\n\\end{array}\\right.\\), continuous on R. [May ’15 (AP), ’11]
\nSolution:
\nWe find the limit at a = 1
\n\"TS
\n\u2234 f is continuous at x = 1
\nHence f is continuous on R.<\/p>\n

Question 2.
\nIs 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]
\nSolution:
\nGiven, f(x) = \\(\\begin{cases}\\frac{\\sin 2 x}{x} & \\text { if } x \\neq 0 \\\\ 1, & \\text { if } x=0\\end{cases}\\)
\nTake a = 0
\n\"TS
\n\u2234 f is discontinuous at x = 0.<\/p>\n

\"TS<\/p>\n

Question 3.
\nCheck the continuity of the following function at 2.
\nf(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]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 4.
\nCheck 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]
\nSolution:
\n\"TS<\/p>\n

Question 5.
\nProve that the functions sin x and cos x are continuous on R. [May ’08]
\nSolution:
\n(i) Let f(x) = sin x and a \u2208 R
\n\\(\\lim _{x \\rightarrow a} f(x)=\\lim _{x \\rightarrow a} \\sin x\\) = sin a = f(a)
\n\u2234 \\(\\lim _{x \\rightarrow a} f(x)\\) = f(a)
\n\u2234 f is continuous at x = a
\n\u2234 Since a is arbitrary, f is continuous on R.
\n(ii) Let g(x) = cos x and a \u2208 R
\n\\(\\lim _{x \\rightarrow a} g(x)=\\lim _{x \\rightarrow a} \\cos x\\) = cos a = g(a)
\n\u2234 \\(\\lim _{x \\rightarrow a} g(x)\\) = g(a)
\n\u2234 g is continuous at x = a
\n\u2234 since a is arbitrary, g is continuous on R.<\/p>\n

Question 6.
\nFind 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]
\nSolution:
\nGiven, 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}\\)
\nSince f is continuous on R.
\nf is continuous at 0, 3.
\n\"TS
\nSince f is continuous at x = 3 then
\nLHL = RHL
\n3b + 3 = -3
\n3b = -3 – 3
\nb = -2
\n\u2234 a = 0, b = -2<\/p>\n

\"TS<\/p>\n

Question 7.
\nShow that f(x) = \\(\\left\\{\\begin{array}{cl}
\n\\frac{\\cos a x-\\cos b x}{x^2} & \\text { if } x \\neq 0 \\\\
\n\\frac{1}{2}\\left(b^2-a^2\\right) & \\text { if } x=0
\n\\end{array}\\right.\\) where a and b are real constants, is continuous at ‘0’. [Mar. ’17 (TS), ’13(old), ’09; May ’14; B.P.]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Some More Maths 1B Limits and Continuity Important Questions Short Answer Type<\/h3>\n

Question 8.
\nFind \\(\\lim _{x \\rightarrow 3} \\frac{x^3-6 x^2+x}{x^2-9}\\)
\nSolution:
\n\"TS<\/p>\n

Question 9.
\nFind \\(\\lim _{x \\rightarrow 3} \\frac{x^3-3 x^2}{x^2-5 x+6}\\)
\nSolution:
\n\"TS<\/p>\n

Question 10.
\nCompute \\(\\lim _{x \\rightarrow 3} \\frac{x^4-81}{2 x^2-5 x-3}\\)
\nSolution:
\nGiven, \\(\\lim _{x \\rightarrow 3} \\frac{x^4-81}{2 x^2-5 x-3}\\)
\n\"TS<\/p>\n

Question 11.
\nCompute \\(\\lim _{x \\rightarrow 3} \\frac{x^2-8 x+15}{x^2-9}\\). [Mar. ’16 (AP & TS)]
\nSolution:
\n\"TS<\/p>\n

Question 12.
\nIf f(x) = \\(-\\sqrt{25-x^2}\\) then find \\(\\lim _{x \\rightarrow 1} \\frac{f(x)-f(1)}{x-1}\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 13.
\nCompute \\(\\lim _{x \\rightarrow 0} \\frac{\\sin a x}{\\sin b x}\\), b \u2260 0, a \u2260 b. [Mar. ’18 (TS)]
\nSolution:
\nGiven, \\(\\lim _{x \\rightarrow 0} \\frac{\\sin a x}{\\sin b x}\\)
\nNow dividing the numerator and denominator by x, we get
\n\"TS<\/p>\n

Question 14.
\nCompute \\(\\lim _{x \\rightarrow 0} \\frac{e^{3 x}-1}{x}\\). [Mar. ’18 (AP); May ’15 (TS)]
\nSolution:
\n\"TS<\/p>\n

Question 15.
\nEvaluate \\(\\lim _{x \\rightarrow 1} \\frac{\\log _e x}{x-1}\\).
\nSolution:
\n\"TS<\/p>\n

Question 16.
\nCompute \\(\\lim _{x \\rightarrow 3} \\frac{e^x-e^3}{x-3}\\)
\nSolution:
\n\"TS<\/p>\n

Question 17.
\nCompute \\(\\lim _{x \\rightarrow 0} \\frac{e^{\\sin x}-1}{x}\\)
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 18.
\nCompute \\(\\lim _{x \\rightarrow 1} \\frac{(2 x-1)(\\sqrt{x}-1)}{2 x^2+x-3}\\)
\nSolution:
\n\"TS<\/p>\n

Question 19.
\nCompute \\(\\lim _{x \\rightarrow 0} \\frac{\\log _e(1+5 x)}{x}\\)
\nSolution:
\n\"TS<\/p>\n

Question 20.
\nCompute \\(\\lim _{x \\rightarrow 0} \\frac{(1+x)^{\\frac{1}{8}}-(1-x)^{\\frac{1}{8}}}{x}\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 21.
\nCompute \\(\\lim _{x \\rightarrow 0} \\frac{1-\\cos x}{x}\\)
\nSolution:
\n\"TS<\/p>\n

Question 22.
\nCompute \\(\\lim _{x \\rightarrow 0} \\frac{\\sec x-1}{x^2}\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 23.
\nCompute \\(\\lim _{x \\rightarrow 0} \\frac{1-\\cos m x}{1-\\cos n x}\\), n \u2260 0.
\nSolution:
\n\"TS<\/p>\n

Question 24.
\nCompute \\(\\lim _{x \\rightarrow 0} \\frac{x\\left(e^x-1\\right)}{1-\\cos x}\\). [May ’14]
\nSolution:
\nGiven, \\(\\lim _{x \\rightarrow 0} \\frac{x\\left(e^x-1\\right)}{1-\\cos x}\\)
\n\"TS<\/p>\n

Question 25.
\nCompute \\(\\lim _{x \\rightarrow 0} \\frac{\\log \\left(1+x^3\\right)}{\\sin ^3 x}\\)
\nSolution:
\n\"TS<\/p>\n

Question 26.
\nCompute \\(\\lim _{x \\rightarrow 0} \\frac{x \\tan 2 x-2 x \\tan x}{(1-\\cos 2 x)^2}\\)
\nSolution:
\nGiven, \\(\\lim _{x \\rightarrow 0} \\frac{x \\tan 2 x-2 x \\tan x}{(1-\\cos 2 x)^2}\\)
\n\"TS<\/p>\n

Question 27.
\nCompute \\(\\lim _{x \\rightarrow \\infty} \\frac{x^2-\\sin x}{x^2-2}\\). [May ’16 (AP)]
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 28.
\nCompute \\(\\lim _{x \\rightarrow 3} \\frac{x^2+3 x+2}{x^2-6 x+9}\\). [Mar. ’19 (AP)]
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 29.
\nCompute \\(\\lim _{x \\rightarrow \\infty} \\frac{3 x^2+4 x+5}{2 x^3+3 x-7}\\). [May ’15 (AP)]
\nSolution:
\n\"TS<\/p>\n

Question 30.
\nCompute \\(\\lim _{x \\rightarrow \\infty} \\frac{6 x^2-x+7}{x+3}\\).
\nSolution:
\n\"TS<\/p>\n

Question 31.
\nCompute \\(\\lim _{x \\rightarrow \\infty} \\frac{x^2+5 x+2}{2 x^2-5 x+1}\\). [May ’14; Mar. ’17 (AP)]
\nSolution:
\n\"TS<\/p>\n

Question 32.
\nCompute \\(\\lim _{x \\rightarrow 2}\\left[\\frac{1}{x-2}-\\frac{4}{x^2-4}\\right]\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 33.
\nCompute \\(\\lim _{x \\rightarrow-\\infty} \\frac{5 x^3+4}{\\sqrt{2 x^4+1}}\\)
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 34.
\nCompute \\(\\lim _{x \\rightarrow \\infty} \\frac{2+\\cos ^2 x}{x+2007}\\)
\nSolution:
\nGiven, \\(\\lim _{x \\rightarrow \\infty} \\frac{2+\\cos ^2 x}{x+2007}\\)
\nWe know that
\n-1 \u2264 cos x \u2264 1
\n0 \u2264 cos2<\/sup>x \u2264 1
\n2 + 0 \u2264 2 + cos2<\/sup>x \u2264 2 + 1
\n2 \u2264 2 + cos2<\/sup>x \u2264 3
\n\\(\\frac{2}{x+2007} \\leq \\frac{2+\\cos ^2 x}{x+2007} \\leq \\frac{3}{x+2007}\\)
\n\"TS<\/p>\n

Question 35.
\nCompute \\(\\lim _{x \\rightarrow-\\infty} \\frac{6 x^2-\\cos 3 x}{x^2+5}\\)
\nSolution:
\nWe know that
\n-1 \u2264 cos x \u2264 1
\n-1 \u2265 cos 3x \u2265 1
\n1 \u2265 -cos 3x \u2265 -1
\n-1 \u2264 -cos 3x \u2264 1
\n\"TS<\/p>\n

Question 36.
\nShow that f, given by f(x) = \\(\\frac{\\mathbf{x}-|\\mathbf{x}|}{\\mathbf{x}}\\) (x \u2260 0), is continuous on R – {0}.
\nSolution:
\n\"TS
\n\u2234 f is discontinuous at x = 0.
\n\u2234 Hence f is continuous on R – {0}.<\/p>\n

Question 37.
\nIf 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’.
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 38.
\nIf 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)]
\nSolution:
\nGiven, \\(\\begin{cases}\\mathbf{k}^2 x-k & \\text { if } \\mathbf{k} \\geq 1 \\\\ 2 & \\text { if } x<1\\end{cases}\\)
\n\u2234 f is continuous on R
\n\u2234 f is continuous at x = 1
\nat x = 1, LHL = RHL = f(1) ………(1)
\n\"TS
\nFrom (1), LHL = RHL
\n\u21d2 2 = k2<\/sup> – k
\n\u21d2 k2<\/sup> – k – 2 = 0
\n\u21d2 k2<\/sup> – 2k + k – 2 = 0
\n\u21d2 k(k – 2) + 1(k – 2) = 0
\n\u21d2 (k – 2)(k + 1) = 0
\n\u21d2 k – 2 = 0 (or) k + 1 = 0
\n\u21d2 k = 2 (or) k = -1<\/p>\n

Question 39.
\nCheck the continuity of ‘f’ given by f(x) = \\(\\left\\{\\begin{array}{rlr}
\n4-x^2, & \\text { if } & x \\leq 0 \\\\
\n\\mathbf{x}-5, & \\text { if } & 0 4 x^2-9, & \\text { if } & 1<x<2 \\\\
\n3 x+4, & \\text { if } & x \\geq 2
\n\\end{array}\\right.\\) at points x = 0, 1, 2. [Mar. ’16 (TS)]
\nSolution:
\n\"TS
\n\"TS<\/p>\n","protected":false},"excerpt":{"rendered":"

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) = , continuous … Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/4235"}],"collection":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/comments?post=4235"}],"version-history":[{"count":5,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/4235\/revisions"}],"predecessor-version":[{"id":5711,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/4235\/revisions\/5711"}],"wp:attachment":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/media?parent=4235"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/categories?post=4235"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/tags?post=4235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}