Limits/Annuties

[TheNextOne]

Hi everyone, I am new to the forum and just had some questions. Please try to respond ASAP as I have an exam on this material in 2 hours. Thank you.

1. A person purchases an annuity today consisting of equal payments at
the end of every three months for the next three years and the interest rate is 6%
compounded quarterly. He then plans to make no more deposits, but leave the money in the account to accumulate interest. He calculates that five years from now he will have 10, 000.00 in his account. Find, to the nearest cent, what the equal payments intothe annuity fund are.

For this question, I took the present value of 10,000 from years 5 to 3 and got 8877.11. Then, I plugged that into the FV annuity formula and got 680.76 as the quarteryly payment. Is this correct?

2. Lim {(e^-x) -(3e^x)}/[1+(e^x)]
x= + infinity

For this, I got the answer to be negative infinity since the numerator will be negative. Is this correct?

3. Lim {(3x+1)^20 -1}/x
x= 0

I have no idea how to do this. I tried rationalizing but I couldn't work it out.

Thank you.

 

[galactus]

2. Lim {(e^-x) -(3e^x)}/[1+(e^x)]
x= + infinity

For this, I got the answer to be negative infinity since the numerator will be negative. Is this correct?

Here's a little help on this one:

\(\displaystyle \L\\\lim_{x\to\infty}\frac{e^{-x}-3e^{x}}{1+e^{x}}\)

L'Hopital's rule. Differentiate numerator and denominator:

\(\displaystyle \L\\\lim_{x\to\infty}\frac{-e^{-x}-3e^{x}}{e^{x}}\)

\(\displaystyle \L\\\lim_{x\to\infty}\frac{{-}e^{-x}}{e^{x}}-\frac{3e^{x}}{e^{x}}\)

\(\displaystyle \L\\\lim_{x\to\infty}{-}e^{-2x}-3\)

=-3

For your last limit, do the same thing

Differentiate top and bottom:

\(\displaystyle \L\\\lim_{x\to\infty}\frac{60(3x+1)^{19}}{1}=60(3x+1)^{19}\)

It's easy to see, as x approaches 0, what the limit is. Isn't it?.

 

[tkhunny]

For this question, I took the present value of 10,000 from years 5 to 3 and got 8877.11. Then, I plugged that into the FV annuity formula and got 680.76 as the quarteryly payment.

Must be some rounding problem in there, somwhere. I get $680.70. Check your intermediate results and DON'T round them.