# Math Help - One sided limit question

1. ## One sided limit question

$
\frac{12sin^3x}{12x^3}
$

$
\lim x \to 0^-
$

Heres what I have...
can be rewritten as:
$
\frac{12(sinx)^3}{12x^3}=\frac{(sinx)^3}{x^3}=\fra c{sinx}{x}*\frac{sinx}{x}*\frac{sinx}{x}=1*1*1=1
$

Up to this point, am I correct? How does the fact that it wants the limit as x approaches zero from the left? I only know how to work with these one-sided limits in a peicewise function. What do I do from here & what do I need to learn to solve one sided limits that are NOT in a peicewise function?

Thanks!

2. Originally Posted by Vamz
$
\frac{12sin^3x}{12x^3}
$

$
\lim x \to 0^-
$

Heres what I have...
can be rewritten as:
$
\frac{12(sinx)^3}{12x^3}=\frac{(sinx)^3}{x^3}=\fra c{sinx}{x}*\frac{sinx}{x}*\frac{sinx}{x}=1*1*1=1
$

Up to this point, am I correct? How does the fact that it wants the limit as x approaches zero from the left? I only know how to work with these one-sided limits in a peicewise function. What do I do from here & what do I need to learn to solve one sided limits that are NOT in a peicewise function?

Thanks!
$\displaystyle \lim_{x \to 0^-} \frac{12 \sin^3 (x)}{12 x^3} = \lim_{x \to 0^+} \frac{12 \sin^3 (x)}{12 x^3} = \lim_{x \to 0} \frac{12 \sin^3 (x)}{12 x^3} = 1$.

3. Originally Posted by Vamz
$
\frac{12sin^3x}{12x^3}
$

$
\lim x \to 0^-
$

Heres what I have...
can be rewritten as:

$
\frac{12(sinx)^3}{12x^3}=\frac{(sinx)^3}{x^3}=\fra c{sinx}{x}*\frac{sinx}{x}*\frac{sinx}{x}=1*1*1=1
$

Up to this point, am I correct? How does the fact that it wants the limit as x approaches zero from the left? I only know how to work with these one-sided limits in a peicewise function. What do I do from here & what do I need to learn to solve one sided limits that are NOT in a piecewise function?

Thanks!
$x=-u$

$x\rightarrow\ 0^-\Rightarrow\ u\rightarrow\ 0^+$

$Sinx=Sin(-u)=-Sinu$

$\displaystyle\frac{Sinx}{x}=\frac{-Sinu}{-u}=\frac{Sinu}{u}$