Solve Limit l'Hopital's Homework: e^r

[whatlifeforme]

Homework Statement

evaluate the limit

Homework Equations

lim k->∞ (1+r/k)^k

The Attempt at a Solution

1. lim k->∞ ln(1+rk^-1) / (k^-1)

not sure how to get to this next step:

2. lim k->∞ (-rk^-2)/(1+rk^-1) / -k^-2

not sure how to get to this next step:

3. lim k->∞ r/(1+rk^-1)

not sure how to get to this next step:

4. lim k->∞ rk/k+r

not sure how to get to this next step:

5. lim k->∞ r/1 = r

6. lim k->∞ (1+r/k)^k

7. lim k->∞ f(k) = lim k->∞ e^lnf(k) = e^r

 

[Dick]

Homework Statement

evaluate the limit

Homework Equations

lim k->∞ (1+r/k)^k

The Attempt at a Solution

1. lim k->∞ ln(1+rk^-1) / (k^-1)

not sure how to get to this next step:

2. lim k->∞ (-rk^-2)/(1+rk^-1) / -k^-2

not sure how to get to this next step:

3. lim k->∞ r/(1+rk^-1)

not sure how to get to this next step:

4. lim k->∞ rk/k+r

not sure how to get to this next step:

5. lim k->∞ r/1 = r

6. lim k->∞ (1+r/k)^k

7. lim k->∞ f(k) = lim k->∞ e^lnf(k) = e^r

After you take the log take the derivative of the numerator and denominator, like you usually do with l'Hopital. Use the chain rule. What do you get?

 

[LCKurtz]

Homework Statement

evaluate the limit

Homework Equations

lim k->∞ (1+r/k)^k

The Attempt at a Solution

1. lim k->∞ ln(1+rk^-1) / (k^-1)

not sure how to get to this next step:

2. lim k->∞ (-rk^-2)/(1+rk^-1) / -k^-2

not sure how to get to this next step:

3. lim k->∞ r/(1+rk^-1)

not sure how to get to this next step:

4. lim k->∞ rk/k+r

not sure how to get to this next step:

5. lim k->∞ r/1 = r

6. lim k->∞ (1+r/k)^k

7. lim k->∞ f(k) = lim k->∞ e^lnf(k) = e^r

If you don't understand each step, how did you do each step? Or are you trying to follow someone else's work? Do you have the limit definition of $e$ to work with?$$
e = lim_{n\rightarrow \infty}\left(1 + \frac 1 n\right)^n$$If so, there is no need for L'Hospital's rule. Try the substitution $k = ru$.