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Unformatted text preview: Math 1a L’Hˆopital’s Rule Day Two Fall, 2009 ’ & $ % More Indeterminate Forms Now let’s consider several new indeterminate forms: Type 0 : lim x → a f ( x ) g ( x ) with lim x → a f ( x ) = 0 and lim x → a g ( x ) = 0 Type ∞ : lim x → a f ( x ) g ( x ) with lim x → a f ( x ) = ∞ and lim x → a g ( x ) = 0 Type 1 ∞ : lim x → a f ( x ) g ( x ) with lim x → a f ( x ) = 1 and lim x → a g ( x ) = ∞ or ∞ 1 Verify that each of these limits is in one of our new indeterminate forms, then use algebra to rewrite it as an indeterminate form of type or type ∞ ∞ . Then find the limit using L’Hˆ opital’s Rule. (a) lim x → (sin( x )) x (b) lim x → (1 3 x ) 1 /x (c) lim x →∞ x 1 /x (d) lim x →∞ ( x ln( x )) 2 One of the standard definitions of the letter e is as a limit: e = lim x →∞ 1 + 1 x x . Rewrite this in a form in which we can use L’Hˆ opital’s Rule, and use this to verify that the limit is, in fact, e . 3 When we study compounded interest, we get the factor ( 1 + r m ) mt for an interest rate r , compounded m times per year for t years. Increasing m naturally increases this factor, but does it grow to infinity? Use the technique of the previous problem to compute lim m →∞ 1 + r m mt , the multiplier for socalled continuous compounding . Math 1a Review Problems Fall, 2009 4 Let v 1 be the velocity of light in air and v 2 the velocity of light in water. According to Fermat’s principle, a ray of light will travel from a point A to a point B in the water by a path ACB that minimizes the time taken. Show that sin( θ 1 ) sin( θ 2 ) = v 1 v 2 , where θ 1 (the angle of incidence) and θ 2 (the angle of refraction) are as shown. This equation is known as Snell’s Law . Hint: Let x be the distance DC , and w the distance DE . What x minimizes the time taken? .................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
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 Fall '09
 BenedictGross
 Math, ........., Limit of a function, Exponentiation, Indeterminate form

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