# Limits

#### Esthephane

Need helps solving these. I have no clue how to do them. (Thinking)   #### matheagle

MHF Hall of Honor
The first one is a zero times an infinity, so put the sine x in the denominator and make it infinity over infinity.

Actually I would just use sin x over x goes to 1...

$$\displaystyle \left({\sin x\over x}\right)\left({\ln 4+\ln x\over x^{-1}}\right)\sim -x\to 0$$

The second one uses conjugates.

$$\displaystyle \sqrt{x^2+3x}-x = {(\sqrt{x^2+3x}-x)(\sqrt{x^2+3x}+x)\over \sqrt{x^2+3x}+x}$$

$$\displaystyle = {3x\over \sqrt{x^2+3x}+x}$$

Now divide by x and take your limit, but be careful when passing the x though the square root.
3/2

The third one is $$\displaystyle e^{-4}$$ by a theorem.
But it's easy to prove by taking the logarithm and using Lopies' Rule

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#### CaptainBlack

MHF Hall of Fame
Need helps solving these. I have no clue how to do them. (Thinking) rewrite:

$$\displaystyle \lim_{x\to 0^+} \sin(x) \ln(4x)=\lim_{x\to 0^+} \frac{\ln(4x)}{[1/\sin(x)]}$$

and apply L'Hopitals's rule.

CB

#### Random Variable

rewrite:

$$\displaystyle \lim_{x\to 0^+} \sin(x) \ln(4x)=\lim_{x\to 0^+} \frac{\ln(4x)}{[1/\sin(x)]}$$

and apply L'Hopitals's rule.

CB
twice

#### CaptainBlack

MHF Hall of Fame For this you need to know that:

$$\displaystyle e^z=\lim_{n\to \infty}\left( 1+\frac{z}{n}\right)^n$$

and put $$\displaystyle x=1/n$$

CB

#### CaptainBlack

MHF Hall of Fame
No once and know that:

$$\displaystyle \lim_{x \to 0} \frac{\sin(x)}{x}=1$$

CB

• Random Variable

#### Random Variable

No once and know that:

$$\displaystyle \lim_{x \to 0} \frac{\sin(x)}{x}=1$$

CB
Yeah, that will work, too.

For the last limit, you could also take the log and apply L'Hospital's rule.

#### matheagle

MHF Hall of Honor
I used sin x over x immediately.