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.