A couple of limit questions: $\lim{x \to 2^-} \frac{3x^2 + x - 7}{|x - 2|}$ and $\lim_{x \to 0^+} \frac{x}{|x|}$

Question

I'm working on my calculus homework right now and have become stumped on two questions about limits.

1. $\lim\limits_{x \to 2^-} \frac{3x^2 + x - 7}{|x - 2|}$
2. $\lim\limits_{x \to 0^+} \frac{x}{|x|}$

For the first part, plugging in doesn't work as it will give you 7/0. And you cannot factor it, so I'm at a complete loss.

As for the second question I didn't get very far either, as just plugging in would give you 0/0.

Can someone help me out with this? Also, I'm sorry if this isn't in the proper format for posting equations and if it's incorrect I'd appreciate it if someone could edit it for me.

Answer 1

Since, in first expression, plugging the value gives result $\frac{7}{0}$ which shows the limit is infinite.

For second part,

$\lim_{x\to0^+}\frac{x}{|x|}=\lim_{x\to0^+}\frac{x}{x}=1$ (I have taken $|x|=x$ as x is approaching $0$ from positive side)

you can read from here : http://www.themathpage.com/acalc/infinity.htm