tutorial 2 solution - NATIONAL UNIVERSITY OF SINGAPORE...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 2 (Solution Notes) 1. Use L’Hopital’s rule to fnd the Following limits. (a) lim x π/ 2 1 sin x 1+cos2 x (b) lim x 0 ln(cos ax ) ln(cos bx ) , a, b > 0 (c) lim x →∞ x tan 1 x (d) lim x 0+ x a ln x , a> 0 (e) lim x 1 x 1 1 x (F) lim x 0 + x sin x (g) lim x 0 ± sin x x ² 1 x 2 Preliminaries The Limit Laws IF L , M , c and k are real numbers and lim x c f ( x )= L and lim x c g ( x M, then 1. Sum Rule : lim x c [ f ( x )+ g ( x )] = L + M The limit oF the sum oF two Functions is the sum oF their limits. 6 2. Diference Rule : lim x c [ f ( x ) g ( x L M The limit oF the di±erence oF two Functions is the di±erence oF their limits. 7 3. Product Rule : lim x c [ f ( x ) · g ( x L · M The limit oF a product oF two Functions is the product oF their limits. 8 4. Constant Multiple Rule : lim x c [ k · f ( x k · L The limit oF a constant times a Function is the constant times the limit oF the Function. 6 If L + M is of the form +( −∞ ), then this rule does not hold any more, and you need to consider what is f ( x g ( x ). 7 If L M is of the form ∞−∞ , then this rule does not hold any more, and you need to consider what is f ( x ) g ( x ). 8 If L · M is of the form 0 ·∞ , then you need to use L’Hopital’s rule instead. 16
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5. Quotient Rule : lim x c [ f ( x ) g ( x ) ]= L M The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero. 9 6. Power Rule :I f r and s are integers with no common factor and s ± =0then lim x c [ f ( x )] r/s = L r/s provided that L r/s is a real number. (If s is even, we assume that L> 0.) The limit of a rational power of a function is that power of the limit of the function, provided the latter is a real number. L’Hopital’s Rule The rule is named after the 17th-century French mathematician Guillaume de L’Hopital, who published the rule in his book l’Analyse des In±niment Petits pour l’Intelligence des Lignes Courbes (literal translation: Analysis of the In±nitely Small to Understand Curved Lines) (1696), the ±rst book about di²erential calculus. L’Hospital’s Rule Suppose and are differentiable and near (except possibly at ). Suppose that and or that and (In other words, we have an indeterminate form of type or .) Then if the limit on the right side exists (or is or ). ±² ² lim x l a f ± x ² t ± x ² ± lim x l a f ³ ± x ² t ³ ± x ² ² ³ ² 0 0 lim x l a t ± x ² ± ´² lim x l a f ± x ² ± ´² lim x l a t ± x ² ± 0 lim x l a f ± x ² ± 0 a a t ³ ± x ² ² 0 t f Remarks: L’Hospital’s Rule is also valid for one-sided limits and for limits at in±nity or negative in±nity; that is, “ x a ”can be replaced by any of the symbols x a + , x a , x →∞ ,or x →−∞ .
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This note was uploaded on 05/05/2009 for the course MA MA1505 taught by Professor Ma1505 during the Spring '09 term at National University of Juridical Sciences.

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tutorial 2 solution - NATIONAL UNIVERSITY OF SINGAPORE...

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