{"id":415,"date":"2017-09-07T12:21:06","date_gmt":"2017-09-07T06:51:06","guid":{"rendered":"http:\/\/ibelitetutor.com\/blog\/?p=415"},"modified":"2025-05-27T00:40:18","modified_gmt":"2025-05-26T19:10:18","slug":"limit-problems","status":"publish","type":"post","link":"https:\/\/ibelitetutor.com\/blog\/limit-problems\/","title":{"rendered":"How To Solve Limit Problems"},"content":{"rendered":"Limit problems’ first post by\u00a0Online IB Tutors<\/a><\/strong> was all about some basic as well as advanced concepts of limits problems. Here Our IB Maths Tutors <\/strong>shall discuss different methods to solve Limit questions.\r\n

How To Solve Limit Problems<\/span><\/h2>\r\nBased on the type of function, we can divide all our work into sections-:<\/span>\r\n\r\nAlgebraic\u00a0Limit Problems- <\/strong>Problems that involve algebraic functions are called algebraic limits. They can be further divided into the following sections:-\r\n\r\nDirect Substitution Method –<\/strong>Suppose we have to find.\u00a0\u00a0we can directly substitute the value of the limit of the variable (i.e replace x=a) in the expression.\r\n\r\n\u25ba\u00a0<\/strong>If f(a) is finite then L=f(a)\r\n\r\n\u25ba\u00a0<\/strong>If f(a) is undefined then L doesn’t exist\r\n\r\n\u25ba\u00a0<\/strong>If f(a) is indeterminate \u00a0then this method fails\r\n\r\n \r\n\r\nExample-1:- <\/strong>Find value of \u00a0<\/strong>(x\u00b2-5x+6)\r\n\r\n \r\n\r\nAns:\u00a0 \u00a0(x\u00b2-5x+6)<\/strong>\r\n\r\n=2\u00b2-5.2+6\r\n\r\n=4-10+6\r\n\r\n=0\r\n\r\nFactorization Method –<\/strong> Suppose we need to find \u00a0 \u00a0 \u00a0where P(x) and Q(x) are polynomials, then we factorize both P(x) and Q(x) in their lowest form. Then we simplify the given expression as much as possible. After all this, we put the limit. We try to ensure that we don’t get zero in the denominator.\r\n\r\nwe can use these tricks-\r\n\r\n\u25ba\u00a0<\/strong>\r\n\r\nwhere n can be even or odd positive integer\r\n\r\n \u25ba\u00a0<\/strong>\"{x^n}\r\n\r\n\r\n\r\nwhere n is an odd positive integer. This formula is not applicable when n is even\r\n\r\n\u25ba <\/strong>Sometimes, we can directly use the below formula to evaluate the limit<\/strong>\r\n\r\n \r\n\r\n\u25ba<\/strong>If the degree of the numerator is more than or equal to the degree of the denominator, then we should divide.\r\n\r\nExample-2:- <\/strong>Evaluate\r\n\r\n \r\n\r\nAns:\r\n\r\n\r\n\r\n \r\n\r\n= 10\r\n\r\n \r\n\r\n\r\n\r\nRationalization Method:- <\/strong>If we ever get 0\/0 form\u00a0in the problems involving square roots, then there must be a common factor in both numerator and denominator which must be canceled out to get a meaningful form. To cancel this common factor, we rationalize the denominator or numerator or both.\r\n\r\nExample-3:-<\/strong>Evaluate\r\n\r\n \r\n\r\nAns-\r\n\r\n\r\n\r\nRationalizing the denominator and the numerator both\r\n\r\n \r\n\r\n= \u00a03.\u00a0\"\r\n\r\n \r\n\r\n= 3. \u00a0\"\\left(\r\n\r\n\r\n\r\n \r\n\r\n=\u00a0\"\\frac{{3\r\n

Solving of not-defined type Limit problems<\/strong><\/h4>\r\n\u00a0<\/strong>If we are given a problem with \"\u00a0we first of all note the highest st power of x in the whole question. After this, we divide both numerator and denominator by that power. This will convert both numerator and denominator into 1\/x form. After this, we can use the following formula\r\n\r\n\"\r\n\r\nwe can use the below result to solve most problems of this category\r\n\r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\"\r\n\r\n\r\n\r\n\u25ba If m=n then our answer is \u00a0\u00a0\"{\\frac{{a{}_0}}{{b{}_0}}}\"\r\n\r\n\u25ba If m<n then our answer is \u00a00\r\n\r\n\r\n\u25ba If m>n then our answer is \u00a0not defined\r\n\r\n\r\n \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\nBased On Logarithmic\u00a0and Exponential Functions:-<\/strong> There are several short cut tricks that we can use to solve these types of problems. A few of them are as follows-\r\n\r\n 1. \u00a0\u00a0\"\r\n\r\n2.\u00a0\"\r\n\r\n3. \u00a0\"\r\n\r\n4.\u00a0\"\r\n\r\n5. If\u00a0\"\u00a0and \u00a0\"\u00a0 \u00a0then \u00a0\u00a0\"\r\n\r\n6. \u00a0\"\r\n\r\n7.\u00a0\"\r\n\r\n8.\u00a0\"\r\n\r\nExample-3:- Evaluate \u00a0 \u00a0<\/strong>\"\r\n\r\n \r\n\r\nAns:- \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\"\r\n\r\n \r\n\r\n= \u00a0\u00a0\"\r\n\r\n \r\n\r\n= \u00a0\u00a0\"\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0(by rule viii)\r\n\r\n \r\n\r\n= \u00a0\"{e^0}\r\n\r\n \r\n\r\n=1\r\n\r\n\r\n\r\n \r\n\r\nProblems Based on Series:-<\/strong>\r\n\r\n(i)<\/strong>\u00a0\u00a0ex<\/sup>\u00a0=1+x\/1!+x3<\/sup>\/3!+x4<\/sup>\/4!\u2026\u2026\"\\infty<\/strong>\r\n\r\n \r\n\r\n(ii) \u00a0ax<\/sup>=1+(xloga)\/1!+ (xloga)2<\/sup>\/2!+ (xloga)3<\/sup>\/3!+ (xloga)4<\/sup>\/4!+\u2026\u2026\u2026.<\/strong>where a > 0\r\n\r\n \r\n\r\n(iii)\u00a0 \u00a0ln(1-x)=x-x2<\/sup>\/2+x3<\/sup>\/3-x4<\/sup>\/4\u2026\u2026\u2026.\u00a0 \u00a0 where -1 < x \u00a01<\/strong>\r\n\r\n \r\n\r\n(iv)\u00a0\u00a0ln(1-x)=-x-x2<\/sup>\/2-x3<\/sup>\/3-x4<\/sup>\/4\u2026\u2026\u2026.\u00a0 \u00a0\u00a0 where\u00a0 -1 x < 1<\/strong>\r\n\r\n \r\n\r\n(v ) \u00a0\"\\sin<\/strong>\r\n\r\n \r\n\r\n(vi)\u00a0\"\\cos<\/strong>\r\n\r\n \r\n\r\n(v)<\/strong>\u00a0\u00a0\"\\tan\r\n\r\n \r\n\r\n(vi) \u00a0\"{\\sin<\/strong>\r\n\r\n(vii<\/strong>)\u00a0\"{\\tan\r\n\r\n \r\n\r\n(viii)\u00a0<\/strong>\"{(1\r\n\r\nExamples:- \u00a0 Find Value of \u00a0 \u00a0<\/strong>\"\r\n\r\n \r\n\r\nAns:- \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\" \r\n\r\n\r\n\r\n=\u00a0\" \r\n\r\n\r\n\r\n \r\n\r\n=\u00a0\"\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n= \u00a0\" \r\n\r\n \r\n\r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 = \u00a0\u00a0\" \r\n\r\n \r\n\r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0= \u00a0 \u00a0\"\\frac{{ \r\n\r\n\r\n\r\n \r\n\r\n= \u00a0 \u00a0\"\\frac{{ \r\n\r\n\r\n\r\n \r\n\r\n= \u00a0 \u00a0\"\\frac{{ \r\n\r\nLimits Based on Trigonometric Functions:-<\/strong>\r\n\r\n\"\r\n\r\nThe above concept is used to solve limit problems involving trigonometric functions. We also use substitution, factorization, rationalization and other algebraic methods to evaluate these types of problems.\r\n\r\nExample:-\r\n\r\nExample 4:<\/strong>\u00a0Evaluate.\"\"\r\n\r\nAns:- \u00a0 \u00a0 \u00a0 Because sec\u00a0x<\/em>\u00a0= 1\/cos\u00a0x<\/em>, you find that\r\n\r\nL’ Hospital Rule:-\u00a0<\/strong>L’H\u00f4pital’s rule\u00a0or\u00a0L’Hospital’s rule\u00a0(French:\u00a0<\/small>[lopital] was first introduced by\u00a0\u00a0French\u00a0mathematician\u00a0Guillaume de l’H\u00f4pital in his 1696 treatise, this is supposed to be the first book on Differential calculus. But the rule was originally discovered in 1694 by\u00a0\u00a0Johann Bernoulli who was a Swiss Mathematician.<\/span>\r\n\r\nIf f(x) and g(x) be two functions in such a way that\r\n\r\n(i) \u00a0\"\u00a0or \u00a0\"\r\n\r\n \r\n\r\n(ii) both are continuous at x=a\r\n\r\n \r\n\r\n(iii) both are differentiable at x=a\r\n\r\n \r\n\r\n(iv) f'(x) and g'(x) are both continuous at x=a then\r\n\r\n \r\n\r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\"\r\n\r\nIf the result is still in 0\/0 or in the \"\\frac{\\inftyform we can again differentiate and write like this\r\n\r\n \r\n\r\n\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\"\r\n\r\nYou can download this PDF to get Practice Questions. You can put your doubts in the comment section for a quick reply.\r\n\r\nHire the best online maths tutors or the best ib tutors in Delhi\r\n\r\n\"ib\r\n\r\n\n
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    Limit problems’ first post by\u00a0Online IB Tutors was all about some basic as well as advanced concepts of limits problems. Here Our IB Maths Tutors […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[],"class_list":["post-415","post","type-post","status-publish","format-standard","hentry","category-cbse-tutors"],"_links":{"self":[{"href":"https:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts\/415","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/comments?post=415"}],"version-history":[{"count":1,"href":"https:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts\/415\/revisions"}],"predecessor-version":[{"id":1143,"href":"https:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts\/415\/revisions\/1143"}],"wp:attachment":[{"href":"https:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/media?parent=415"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/categories?post=415"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/tags?post=415"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}