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tutorial 2 solution

# 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 find the following limits. (a) lim x π/ 2 1 sin x 1 + cos 2 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 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. Difference Rule : lim x c [ f ( x ) g ( x ) ] = L M The limit of the difference of two functions is the difference 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 : If r and s are integers with no common factor and s = 0 then 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 Infiniment Petits pour l’Intelligence des Lignes Courbes (literal translation: Analysis of the Infinitely Small to Understand Curved Lines) (1696), the first book about differential 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 ). lim x l a f x t x lim x l a f x t x 0 0 lim x l a t x lim x l a f x lim x l a t x 0 lim x l a f x 0 a a t x 0 t f Remarks: L’Hospital’s Rule is also valid for one-sided limits and for limits at infinity or negative infinity; that is, “ x a ”can be replaced by any of the symbols x a + , x a , x → ∞ , or x → −∞ . L’Hospital’s Rule says that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives, provided that the given conditions are satisfied. It is especially important to verify the conditions regarding the limits of and before using L’Hospital’s Rule.
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tutorial 2 solution - NATIONAL UNIVERSITY OF SINGAPORE...

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