26-LHopital-and-Review

26-LHopital-and-Review - Math 1a L’Hˆopital’s Rule Day...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 re-write 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 . Re-write 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 so-called 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? .................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
View Full Document

Page1 / 4

26-LHopital-and-Review - Math 1a L’Hˆopital’s Rule Day...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online