1. ## Limit of Sequences

Having trouble with this exercise:

Find the limit of the following sequence or determine that the limit does not exist.

$$\bigg\{\bigg(1 + \frac{22}{n}\bigg)^n\bigg\}$$

The problem says to use L'Hôpital's Rule and then simplify the expression and I am getting tripped up by the division. Could someone tell me how they are going from:

$$\lim_{x\to\infty} x ln(1 + \frac{22}{x})$$

After L'Hôpital's Rule:

$$\lim_{x\to\infty} \frac{\frac{-22}{x^2+22x}}{\frac{-1}{x^2}}$$

to

$$\lim_{x\to\infty} \frac{22}{1+\frac{22}{x}}$$

As far as I can tell the largest term common to both the numerator an denominator is $$\frac{1}{x^2}$$ so if I divide them both by it I get this:

$$\frac{\frac{-22x^2}{x^2+22x}}{-1} = \frac{-22+22x}{-1} = 22 - 22x$$

... however in the example they go straight to 22 from $$\lim_{x\to\infty} \frac{22}{1+\frac{22}{x}}$$

In which places am I going wrong? My best guess would be the largest common term but I don't see what else it could be.

2. ## Re: Limit of Sequences

Originally Posted by SDF
Having trouble with this exercise:

Find the limit of the following sequence or determine that the limit does not exist.
$\displaystyle\large{\lim _{x \to \infty }}{\left( {1 + \frac{a}{{x + b}}} \right)^{cx}} = {e^{ac}}$

3. ## Re: Limit of Sequences

Is that supposed to be helpful?

@SDF: as $x \to \infty$ you should have $\frac{22}{x} \to 0$

4. ## Re: Limit of Sequences

Originally Posted by Archie
Is that supposed to be helpful?]
Would you know if it were? Any calculus III students should be able to see at once the limit is $e^{22}$.
It is a waste of time to have to use a substitution on every problem.

5. ## Re: Limit of Sequences

Yes, I would. I also read the part of the OP that said
The problem says to use L'Hôpital's Rule and then simplify the expression.
Did you? Or is reading the question also a waste of time?

6. ## Re: Limit of Sequences

\begin{align*} \frac{\frac{-22}{x^2+22x}}{\frac{-1}{x^2}} &= \frac{\frac{-22x^2}{x^2+22x}}{-1} &(\text{multiplying numerator and denominator by }x^2) \\ &= \frac{22x^2}{x^2+22x} & (\text{simplifying}) \\ &= \frac{22}{1 + \frac{22}{x}} & (\text{dividing numerator and denominator by }x^2) \end{align*}

7. ## Re: Limit of Sequences

Originally Posted by Archie
Yes, I would. I also read the part of the OP that said
Did you? Or is reading the question also a waste of time?
Of course I read it. After reading it is clear that having to do and thing beyond recognizing the form is the waste of time.

8. ## Re: Limit of Sequences

Originally Posted by SDF

The problem says to use L'Hôpital's Rule and then simplify the expression and I am getting tripped up by the division. Could someone tell me how they are going from:

$$\lim_{x\to\infty} x ln(1 + \frac{22}{x})$$

After L'Hôpital's Rule:

$$\lim_{x\to\infty} \frac{\frac{-22}{x^2+22x}}{\frac{-1}{x^2}}$$

to

$$\lim_{x\to\infty} \frac{22}{1+\frac{22}{x}}$$
$\dfrac{ \ \frac{-22}{x^2 + 22x} \ }{(\frac{-1}{x^2})} =$

$\dfrac{-22}{x^2 + 22x}\cdot\ \dfrac{x^2}{-1} \ =$

$\dfrac{22x^2}{x(x + 22)} \ =$

$\dfrac{22x}{x + 22} \ =$

$\dfrac{(\tfrac{1}{x})22x}{(\tfrac{1}{x})(x + 22)} \ =$

$\dfrac{22}{1 + \tfrac{22}{x}}$

9. ## Re: Limit of Sequences

Originally Posted by Plato
Of course I read it. After reading it is clear that having to do and thing beyond recognizing the form is the waste of time.
You don't get to impose the method you want to do. You should set up a separate thread if you want to address it with
your method that isn't on the level that the student is working with.

Originally Posted by Plato
Would you know if it were? Any calculus III students should be able to see at once the limit is $e^{22}$.
It is a waste of time to have to use a substitution on every problem.
This isn't a Calculus III problem. You made a comment to an imaginary problem. If you would stick to the problem at hand,

10. ## Re: Limit of Sequences

Originally Posted by Plato
Of course I read it. After reading it is clear that having to do and thing beyond recognizing the form is the waste of time.
Imagine for a moment that a student of yours submitted homework or an exam answer that ignored the problem statement and derived an answer according to a formula that had been gifted on them from above.