1. ## Help Finding Limits!!!

I need help finding the answers to these, I did some of them but are unsure of the answer.

-Use the limit laws to find the limit if it exists:

1. lim x=> 0- square root of -x
i got 0 for this one, but i'm not sure

2. lim x=> 8 absolute value of (x-8)/x-8
i'm not entirely sure on this one

3. lim x=> 0 (1-2x)^3/x
i got zero on this one

4. lim x=> infinity 3e^3x+2e^x-1/e^3x-e^2x+5
no idea on this one

Any help would be a big help, thanks

I need help finding the answers to these, I did some of them but are unsure of the answer.

-Use the limit laws to find the limit if it exists:

1. lim x=> 0- square root of -x
i got 0 for this one, but i'm not sure

2. lim x=> 8 absolute value of (x-8)/x-8
i'm not entirely sure on this one

3. lim x=> 0 (1-2x)^3/x
i got zero on this one

4. lim x=> infinity 3e^3x+2e^x-1/e^3x-e^2x+5
no idea on this one

Any help would be a big help, thanks
You are correct for 1)

For 2) you must split it into two limits. The absoulte value of a function, f(x) is $f(x) \iff f(x) \geq 0$ but $-f(x) \iff f(x) < 0$

So treat it as two cases:
The first case is when x approaches 8 through positive values, when this happens f(x) > 0, hence |f(x)| = f(x).

$\displaystyle \lim_{x \to 8^+} \frac{x-8}{x-8} = \displaystyle \lim_{x \to 8^+} 1 = 1$

Now through negative values, when this happens f(x) <0, hence |f(x)| = -f(x).

$\displaystyle \lim_{x \to 8^-} \frac{8-x}{x-8} = \displaystyle \lim_{x \to 8^-} -1 = -1$

Since these limits are not equal, the limit does not exist!

For 3):

Use the rules $\displaystyle \lim_{x \to a} (f(x))^n = (\displaystyle \lim_{x \to a} f(x))^n$ and the rules $\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\displaystyle \lim_{x \to a} f(x)}{\displaystyle \lim_{x \to a} g(x)}$

This gives:

$\displaystyle \lim_{x \to 0} \frac{(1-2x)^3}{x} = \frac{(\displaystyle \lim_{x \to 0} (1-2x))^3}{\displaystyle \lim_{x \to 0} (x)} = \frac{1}{0} = \pm \infty$. Limit no existy!

For 4) Is this the equation you meant? $\displaystyle \lim_{x \to \infty} 3e^{3x}+\frac{2e^{x-1}}{e^{3x}}-e^{2x+5}$

If so:

$\displaystyle \lim_{x \to \infty} 3e^{3x}+\frac{2e^{x-1}}{e^{3x}}-e^{2x+5} =\displaystyle \lim_{x \to \infty}(3e^{3x}) + \frac{\displaystyle \lim_{x \to \infty} (2e^{x-1})}{\displaystyle \lim_{x \to \infty} (e^{3x})} -\displaystyle \lim_{x \to \infty} (e^{2x+5})$

$\displaystyle =3 \lim_{x \to \infty}(e^{x})^3 + 2 \frac{\lim_{x \to \infty} (e^x e^{-1})}{\lim_{x \to \infty} (e^{x})^3} -\lim_{x \to \infty} (e^{2x}e^5)$

$\displaystyle =3 \bigg(\lim_{x \to \infty}(e^{x})\bigg)^3 + 2e^{-1} \frac{ \displaystyle \lim_{x \to \infty} (e^x)}{\displaystyle \bigg(\lim_{(x \to \infty} (e^{x})\bigg)^3} -e^5 \displaystyle \lim_{x \to \infty} (e^{x})^2$

$\displaystyle =3 \bigg(\lim_{x \to \infty}(e^{x})\bigg)^3 + 2e^{-1} \frac{ \displaystyle \lim_{x \to \infty} (e^x)}{\displaystyle \bigg(\lim_{x \to \infty} (e^{x})\bigg)^3} -e^5\bigg(\displaystyle \lim_{x \to \infty} (e^{x})\bigg)^2$

Now the only limit you have to handle is $\displaystyle \lim_{x \to \infty} e^x$.