MathExcel: Calculus Among Friends   - by Mike Freeman Sample Worksheet 2 MathExcel, Section 21 Name: _______________________ Worksheet 34 November 30, 1998 Groups should work cooperatively to complete as many of the following as possible. Any questions on problems should be asked by the group, not individuals. L'Hospital's Rule is used for evaluating some limits of the form limx a [f(x)/ g(x)],   limx a f(x) ·g(x),   limx a [f(x)]g(x), and limx a [f(x) - g(x)] and these same limit where ``x a" is replaced by ``x ¥"or ``x -¥". L'Hospital's Rule is applied to these functions only if the answer you get from evaluating the limit is what we call an ``indeterminate form"; that is, of the form 0/0, [(¥)/( ¥)], 0 ·¥, 0 ·( -¥), ¥-¥,    1¥,   1-¥,  ¥0, or 00. Determine which of the following limits gives an ``indeterminate form". If not, evaluate the limit. If so, state which indeterminate form it is. limx -2 [(x-2)/( x+2)] limx 0 [(ex - 1)/ x] limx ¥ [ln(x)/ x] limx 0+ x ln(x) limx ¥ x(1/x) limx ¥ xx limx ¥ x2 - ex limx ¥ ex + cos(x) limx 0+ cos(x)cot(x) Become familiar with the following steps to using L'Hopital's rule to evaluate a limit of the form limx a [f(x)/ g(x)]. Go through the following steps to determine limx 0 [(ex - 1)/( x3)]. Check that lim [f(x)/ g(x)] is an indeterminant form. If not, then L'Hopital's rule cannot be used. Check that f(x) and g(x) are differentiable around the point x = a. (It need not be differentiable at a). If not, L'Hopital's rule cannot be used. Differentiate f and g separately. (DO NOT apply the quotient rule to [f(x)/ g(x)]). Find lim [(f¢(x))/( g¢(x))]. If this limit is finite, +¥, or -¥, then it is equal to lim [f(x)/ g(x)]. If it is also indeterminate, repeat the process. Note: If ``x a" is replaced by ``x ¥", then replace step (b) with the following: (b') Check that f(x) and g(x) are differentiable on some interval (n, ¥) where n is a real number. (You can pick n.) If ``x a" is replaced by ``x -¥", what must be checked in part (b)? Evaluate the following limits: limx -1 [(lnx2)/( 1+x)] limx 0 [tanx/ x] limt 2 [(t2-4)/( t-2)] limx ¥ [(1+(lnx)3)/ x lnx] limx ¥ [(1/x + 2)/( 3/x - 4)] Evaluate the limit   limx 3 [(x - 3)/( x2 - 3)]. Why is L'Hospital's Rule not applicable to this limit? Evaluate the following limits: limx p/2 [(1-sinx)/ cosx] limx 0+ x e1/x (Hint: rewrite as a quotient) limx ¥ (ln(x)) e-x limx p (cscx + cotx) (Hint: find a common denominator) L'Hospital's rule can also be used to evaluate limits in the form of limx a [f(x)]g(x) when it gives an indeterminate form such as 1¥,   1-¥,  ¥0, or 00. The following steps outline how to use L'Hospital Rule in this case. Use the following steps to evaluate limx 0+ (1+x)1/x. Check that limx a [f(x)]g(x) is an indeterminant form. If not, then L'Hopital's rule cannot be used. Let y = [f(x)]g(x). Then ln(y) = g(x) * ln(f(x)). Determine the limit limx a ln(y) = limx a [ln(f(x))/( [1/ g(x)])] using all of the steps of L'Hospital's Rule. (You may have to apply it more than once.) The limit found above is limx a ln(y). Find limx a y using that limx a y = limx a elny = e[ limx a ln(y)]. Evaluate the following limits: limx 0+ (cos(x))csc(x) limx ¥ x1/x Evaluate the following limits: limx ¥ [((lnx)3)/( x2)] limx 0 ( [1/( x4)] - [1/( x2)] ) Evaluate the following limits: limx 2 [(x2-4)/( x-2)] limx [(p)/ 2] [(1-sinx)/ cosx] limx +¥ xln( [(x+1)/( x-1)]). limx [(p)/ 2] sinxsecx limx ¥ limx [(p)/ 2] cos3x sec5x Use l'Hopital's Rule, if appropriate, to find: limx -1 [(lnx2)/( 1+x)] limx 0 [tanx/ x] limt 2 [(t2-4)/( t-2)] limx ¥ [(1+(lnx)3)/ x lnx] limx ¥ [(1/x + 2)/( 3/x - 4)] File translated from TEX by TTH, version 2.25. On 9 Sep 1999, 15:25. Tell me more about this: A general description The MathExcel Leaders and Workshops Some advice and reflections on this plan Why I like this activity Sample Worksheet 1 Sample Worksheet 2 Advantages to MathExcel Comments from MathExcel students More about Mike Freeman

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