{"id":35154,"date":"2022-08-25T10:56:14","date_gmt":"2022-08-25T05:26:14","guid":{"rendered":"https:\/\/apboardsolutions.guru\/?p=35154"},"modified":"2022-08-25T10:56:14","modified_gmt":"2022-08-25T05:26:14","slug":"inter-1st-year-maths-1b-limits-and-continuity-important-questions","status":"publish","type":"post","link":"https:\/\/apboardsolutions.guru\/inter-1st-year-maths-1b-limits-and-continuity-important-questions\/","title":{"rendered":"Inter 1st Year Maths 1B Limits and Continuity Important Questions"},"content":{"rendered":"

Students get through Maths 1B Important Questions<\/a> Inter 1st Year Maths 1B Limits and Continuity Important Questions which are most likely to be asked in the exam.<\/p>\n

Intermediate 1st Year Maths 1B Limits and Continuity Important Questions<\/h2>\n

Question 1.
\nEvaluate \"Inter
\nSolution:
\n\"Inter<\/p>\n

Question 2.
\nCompute \"Inter
\nSolution:
\nWrite f(x) = \\(\\frac{x-2}{x^{3}-8}\\) x \u2260 2 so that
\nf(x) = \\(\\frac{x-2}{x^{3}-8}\\) = \\(\\frac{1}{x^{2}+2 x+4}\\)
\nWrite h(x) = x2<\/sup> + 2x + 4 so that
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 3.
\nFind \"Inter
\nSolution:
\n\"Inter<\/p>\n

Question 4.
\nCompute \"Inter.
\nSolution:
\nFor x \u2260 0, we know that -1 \u2264 sin \\(\\frac{1}{x}\\) \u2264 1
\n\u2234 -x2<\/sup> \u2264 x2<\/sup> . sin \\(\\frac{1}{x}\\) \u2264 x2<\/sup>
\n\"Inter<\/p>\n

Question 5.
\nFind \"Inter.
\nSolution:
\nWe define f : R \u2192 R by f(x) = x2<\/sup> – 5 and g : R \u2192 R by g(x) = 4x + 10.
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 6.
\nFind \"Inter.
\nSolution:
\nWrite F(x) = x3<\/sup> – 6x2<\/sup> + 9x
\n= x(x – 3)2<\/sup> = (x – 3)
\nf(x) where f(x) = x(x – 3)
\nWrite G(x) = x2<\/sup> – 9 = (x – 3) (x + 3)
\n= (x – 3) g(x) where g(x) = x + 3
\n\u2234 \\(\\frac{F(x)}{G(x)}=\\frac{(x-3) f(x)}{(x-3) g(x)}=\\frac{f(x)}{g(x)}\\)
\nand g(3) = 6 \u2260 0.
\nIf F and G are polynomials such that f(x) = (x – a)k<\/sup>, G(x) = (x – a)k<\/sup> g(x) for some k \u2208 N and for some polynomials f(x) and g(x) with
\n\"Inter<\/p>\n

Question 7.
\nFind \"Inter.
\nSolution:
\nWe write F(x) = x3<\/sup> – 3x2<\/sup> = x2<\/sup>(x – 3) = (x – 3)
\nf(x) where f(x) = x2<\/sup>,
\nand G(x) = x2<\/sup> – 5x + 6 = (x – 3)(x – 2)
\n= (x – 3) g(x) where g(x) = x – 2,
\nwith g(3) = 3 – 2 = 1 \u2260 0.
\n\u2234 by applying Theorem g(a) \u2260 0
\n\"Inter<\/p>\n

Question 8.
\nShow that \"Inter and \"Inter (x \u2260 0).
\nSolution:
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 9.
\nLet f : R \u2192 R be defined by
\nf(x) = \\(\\begin{cases}2 x-1 & \\text { if } x<3 \\\\ 5 & \\text { if } x \\geq 3\\end{cases}\\) show that \"Inter.
\nSolution:
\n\"Inter<\/p>\n

Question 10.
\nShow that \"Inter.
\nSolution:
\nObserve that \\(\\sqrt{x^{2}-4}\\) is not defined over (-2, 2)
\n\"Inter<\/p>\n

Question 11.
\n\"Inter
\nSolution:
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 12.
\nShow that \"Inter.
\nSolution:
\n\"Inter<\/p>\n

Question 13.
\nFind \"Inter
\nSolution:
\nFor 0 < |x| < 1, we have
\n\"Inter
\n\"Inter<\/p>\n

Question 14.
\nCompute \"Inter
\nSolution:
\nFor 0 < |x| < 1 \"Inter<\/p>\n

\"Inter<\/p>\n

Question 15.
\nShow that \"Inter
\nSolution:
\n\"Inter<\/p>\n

Question 16.
\nCompute [Mar 13] \"Inter (a > 0, b > 0, b \u2260 1).
\nSolution:
\n\"Inter<\/p>\n

Question 17.
\nCompute \"Inter, b \u2260 0, a \u2260 b.
\nSolution:
\n\"Inter<\/p>\n

Question 18.
\nCompute \"Inter
\nSolution:
\n\"Inter<\/p>\n

Question 19.
\nCompute \"Inter
\nSolution:
\n\"Inter<\/p>\n

Question 20.
\nEvaluate \"Inter
\nSolution:
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 21.
\nShow that \"Inter
\nSolution:
\nGiven \u03b5 > 0, choose \u221e = \\(\\frac{1}{\\sqrt{\\varepsilon}}\\) > 0
\nx > \u221e \u21d2 x > \\(\\frac{1}{\\sqrt{\\varepsilon}}\\) \u21d2 x2<\/sup> > \\(\\frac{1}{\\varepsilon}\\) \u21d2 \\(\\frac{1}{x^{2}}\\) < \u03b5
\n\"Inter<\/p>\n

Question 22.
\nShow that \"Inter
\nSolution:
\nGiven k > 0, let \u221e = log k.
\nx > \u221e \u21d2 ex<\/sup> \u21d2 e\u221e<\/sup> = k
\n\"Inter<\/p>\n

Question 23.
\nCompute \"Inter
\nSolution:
\n\"Inter<\/p>\n

Question 24.
\nEvaluate \"Inter.
\nSolution:
\n\"Inter<\/p>\n

\"Inter<\/p>\n

Question 25.
\nIf f(x) \\(\\frac{a_{n} x^{n}+\\ldots+a_{1} x+a_{0}}{b_{m} x^{m}+\\ldots+b_{1} x+b_{0}}\\) when an<\/sub> > 0, bm<\/sub> > 0, then show that \"Inter f(x) = \u221e if n > m.
\nSolution:
\n\"Inter
\n\"Inter<\/p>\n

Question 26.
\nCompute \"Inter
\nSolution:
\n-1 \u2264 sinx \u2264 1 \u21d2 -1 \u2264 -sinx \u2264 1
\nx2<\/sup> – 1 \u2264 x2<\/sup> – sinx \u2264 x2<\/sup> + 1
\nSince x \u2192 \u221e, suppose that the x2<\/sup> – 2 > 0
\n\"Inter<\/p>\n

Question 27.
\nShow that f(x) = [x] (x \u2208 R is continuous at only those real numbers that are not integers.
\nSolution:
\nCase i) : If a \u2208 z, f(a) = (a) = a
\n\"Inter
\n\u2234 f is not continuous at x = a \u2208 z.<\/p>\n

\"Inter<\/p>\n

Case ii) : If a \u2209 z, then \u2203 n \u2208 z such that n < a < n + 1 then f(a) = (a) = n.
\n\"Inter<\/p>\n

Question 28.
\nIf f : R \u2192 R is such that f(x + y) = f(x) + f(y) for all x, y \u2208 R then f is continuous on R if it is continuous at a single point in R.
\nSolution:
\nLet f be continuous at x0<\/sub> \u2208 R
\n\"Inter
\n\u2234 f is continuous at x.
\nSince x \u2208 R is arbitrary, f is continuous on R.<\/p>\n

Question 29.
\nCheck the continuity of the function f given below at 1 and 2.
\nf(x) = \\(\\left\\{\\begin{array}{cl}
\nx+1 & \\text { if } x \\leq 1 \\\\
\n2 x & \\text { if } 1<x<2 \\\\
\n1+x^{2} & \\text { if } x \\geq 2
\n\\end{array}\\right.\\)
\nSolution:
\n\"Inter
\n\u2234 f is continuous at x = 1
\n\"Inter
\nf is not continuous at x = 2.<\/p>\n

Question 30.
\nShow that the function f defined on R by f(x) = Cos x2<\/sup>, x \u2208 R is continuous function.
\nSolution:
\nWe define h : R \u2192 R by h(x) = x2<\/sup> and
\ng : R \u2192 R by g(x) = cosx.
\nNow, for x \u2208 R
\nhave (goh)(x) = g(h(x)) = g(x2<\/sup>)
\n= cos x2<\/sup> = f(x)
\nSince g and h continuous on their respective domains, by Theorem
\nLet A, B, \u2286 R.
\nLet f : A \u2192 R be continuous on A and let
\ng : B \u2192 R be continuous on B.
\nIf f(A) \u2286 B then the composite function
\ngof : A \u2192 R is continous on A.
\nIt follows that a continuous function on R.<\/p>\n

\"Inter<\/p>\n

Question 31.
\nShow that the function f defined on R by f(x) = |1 + 2x + |x||, x \u2208 R is a continuous function.
\nSolution:
\nWe define g : R \u2192 R by
\ng(x) = 1 + 2x + |x|, x \u2208 R,
\nand h : R \u2192 R by h(x) = |x|, x \u2208 R. Then
\n(hog) (x) = h(g(x)) = h(1 + 2x + |x|)
\n= |1 + 2x + |x|| = f(x).<\/p>\n","protected":false},"excerpt":{"rendered":"

Students get through Maths 1B Important Questions Inter 1st Year Maths 1B Limits and Continuity Important Questions which are most likely to be asked in the exam. Intermediate 1st Year Maths 1B Limits and Continuity Important Questions Question 1. Evaluate Solution: Question 2. Compute Solution: Write f(x) = x \u2260 2 so that f(x) = …<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[15],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35154"}],"collection":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/comments?post=35154"}],"version-history":[{"count":1,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35154\/revisions"}],"predecessor-version":[{"id":35155,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/posts\/35154\/revisions\/35155"}],"wp:attachment":[{"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/media?parent=35154"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/categories?post=35154"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/apboardsolutions.guru\/wp-json\/wp\/v2\/tags?post=35154"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}