F is the function defined by f(x) = (x^3-x)/(x^3-4x)
A) find the limit of f(x), as x approaches 0
B) FInd the zeroes of f(x)
C)Write an equation for the vertical and horizontal asymptotes of the graph of f(x)
D)Describe the symmetry of f(x)
F is the function defined by f(x) = (x^3-x)/(x^3-4x)
A) find the limit of f(x), as x approaches 0
B) FInd the zeroes of f(x)
C)Write an equation for the vertical and horizontal asymptotes of the graph of f(x)
D)Describe the symmetry of f(x)
As Krizalid said, what have you tried?
to get you startedtry factorizing the numerator and denominator before taking the limit. see anything?
for a rational function, the zeroes occur where the numerator is zero, provided the denominator is not zero at the same time.B) FInd the zeroes of f(x)
for rational functions, vertical asymptotes occur where the function is undefined, this does not include "holes" however. do you know what "holes" are?C)Write an equation for the vertical and horizontal asymptotes of the graph of f(x)
as for horizontal asymptotes, find $\displaystyle \lim_{x \to \infty}f(x)$ and $\displaystyle \lim_{x \to - \infty}f(x)$. if either of those are finite, those are your horizontal asymptotes
what types of symmetry are you familiar with? do you know what "odd" and "even" mean when describing functions. is this function either of those? something else?D)Describe the symmetry of f(x)
For A I know how to find the limit using (f(x+h)-f(x))/h but I know there was an easier way to find it using the derivative and the derivative i got was
{[-x^3+x]/[(x^3-4x)^2(3x^2-4)]} + (3x-1)/(x^3-4x)
but I dont know where to go.
Also for D I'm not certain but i think if f(x)=f(-x) its even
and if -f(x)=f(-x) its odd?
Also for asymptotes I know the vertical ones are when the denominator is 0 but what are the ways to find the horizontal ones analytically. I recall that it has something to do with the powers of the numerator and denominator but I'm not quite sure what it is exactly
that is the limit for finding the derivative. the first question has NOTHING to do with that. take my advice for what to do. you simply want $\displaystyle \lim_{x \to 0}f(x)$
ok, so, do either of those fit this function?Also for D I'm not certain but i think if f(x)=f(-x) its even
and if -f(x)=f(-x) its odd?
do you know what kind of symmetry an odd function possesses? an even function?
correct...provided the numerator is not zero at the same time
i told you exactly how to find the horizontal asymptotes. the rules you seem to be referring to can be found here (post #2).but what are the ways to find the horizontal ones analytically. I recall that it has something to do with the powers of the numerator and denominator but I'm not quite sure what it is exactly
so for a) $\displaystyle
\lim_{x \to 0}f(x)
$ $\displaystyle f(x)=-\infty$ ?
a)the zeros are x=1 and x=-1 this is because $\displaystyle \frac{x^3-x}{x^3-4x}=\frac{x(x^2-1)}{x(x^2-4)}=\frac{x^2-1}{x^2-4}$ so $\displaystyle x^2-1=0$, $\displaystyle x=+/- 1$, correct?
b and c were easy, but d is still a little confusing, so far from what i have, it is an even function, correct?
Happy New Year!
I see my mistake, so let me just show all my answer to makes sure the others are correct.
a) as $\displaystyle
\lim_{x \to 0}f(x)
$ $\displaystyle f(x)=\frac{1}{4}$. I got this answer after i followed what Jhevon said and factored out certain terms.
b)$\displaystyle \frac{x^3-x}{x^3-4x}=\frac{x(x^2-1)}{x(x^2-4)}=\frac{x^2-1}{x^2-4}$ so $\displaystyle x^2-1=0$ thus $\displaystyle x=1$, $\displaystyle x=-1$ correct?
c) $\displaystyle x^3-4x=0$ equals $\displaystyle x(x^2-4)=0$ thus zeros at $\displaystyle x=0$ $\displaystyle x=-2$ $\displaystyle x=2$ correct?
d)even because $\displaystyle f(x)=f(-x)$
thanks again for all your wonderful help.
Looks good.
Yes. Since x = 0 is not in the domain, that zero is thrown out.b)$\displaystyle \frac{x^3-x}{x^3-4x}=\frac{x(x^2-1)}{x(x^2-4)}=\frac{x^2-1}{x^2-4}$ so $\displaystyle x^2-1=0$ thus $\displaystyle x=1$, $\displaystyle x=-1$ correct?
Like Jhevon had mentioned, a "hole" is not an asymptote. So only x = 2, x = -2 are the vertical asymptotes. And note that the question also ask for horizontal asymptote.c) $\displaystyle x^3-4x=0$ equals $\displaystyle x(x^2-4)=0$ thus zeros at $\displaystyle x=0$ $\displaystyle x=-2$ $\displaystyle x=2$ correct?
Correct.d)even because $\displaystyle f(x)=f(-x)$