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Sorry I'm not very good at using latex, but here's my shot.

1. Let f(x) = sqrt (1+2x) - 1 - sqrt (x). Find some a where a is positive, such that lim of [tex]\frac{f(x)}{x^a}[/tex] as x approaches 0 from the right is finite and non zero.

I know the problem requires the use of L'Hopital's rule, but I seem to be making circles. When I differentiate, I keep getting indeterminite forms and can't get the limit to be finite

2. A picture 1.4 meters high stands on a wall so that its lower edge is 1.8 meters above the eye of an observer. What is the most favorable distance from the wall for this observer too stand - that is, to maximize his or her angle of vision.

I started by doing tan(theta) = [tex]\frac{1.4+1.8}{x}[/tex] where x is the distance from the wall. To maximize the angle, I took the arctan of each side and differentiated, looking for when it will equal 0, but the derivative is never 0.

3. Evaluate for any fixed number k>0:

lim [tex]\frac{(1^k + 2^k + ... n^k)}{n^(k+1)}[/tex] as n approaches infinity

[Edit: The denominator should be n raised to the quantity of (k+1)]

The numerator looks like a Riemann sum, but I have no idea how to begin solving it.

Thanks for any help

1. Let f(x) = sqrt (1+2x) - 1 - sqrt (x). Find some a where a is positive, such that lim of [tex]\frac{f(x)}{x^a}[/tex] as x approaches 0 from the right is finite and non zero.

I know the problem requires the use of L'Hopital's rule, but I seem to be making circles. When I differentiate, I keep getting indeterminite forms and can't get the limit to be finite

2. A picture 1.4 meters high stands on a wall so that its lower edge is 1.8 meters above the eye of an observer. What is the most favorable distance from the wall for this observer too stand - that is, to maximize his or her angle of vision.

I started by doing tan(theta) = [tex]\frac{1.4+1.8}{x}[/tex] where x is the distance from the wall. To maximize the angle, I took the arctan of each side and differentiated, looking for when it will equal 0, but the derivative is never 0.

3. Evaluate for any fixed number k>0:

lim [tex]\frac{(1^k + 2^k + ... n^k)}{n^(k+1)}[/tex] as n approaches infinity

[Edit: The denominator should be n raised to the quantity of (k+1)]

The numerator looks like a Riemann sum, but I have no idea how to begin solving it.

Thanks for any help

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