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Let g(x) = (x + 4)/(x^2 + x - 12)

 Apr 21, 2020
 #1
avatar+29956 
+1

This graph should help with parts a and b:

 

For part c, note that the denominator can be factored as (x+4)(x-3).

If you just put x = -4 in the original denominator you get zero, but if you put the denominator in factored form, and then cancel the (x+4) term you will have removed the singularity at x = -4. 

 Apr 21, 2020
 #2
avatar+875 
0

I'm still confused about this... How do I use the graph to find the limit of the function? I would like to see work to understand how to find the limit.
I'm sure that the equation of the horizontal asymptote is y = 0 and the equation of the vertical asymptote is x = 3.
By using algebra a coordinate of a removable discontinuity is at x = -4, right? 
 

GAMEMASTERX40  Apr 21, 2020
 #3
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+1

a)
wordy solution:
by looking at that equation, you see that the largest power in the numerator is 1 (x1) and the largest power in the denominator is 2 (x2), since the denominator has the larger power, it will grow faster than the numerator as x gets bigger, so the limit as x approaches infinity of g(x) is 0.

not as wordy solution:
\(lim_{x\rightarrow\infty}\frac{x+4}{x^2+x-12}\\ =lim_{x\rightarrow\infty}\frac{x^2(\frac{1}{x}+\frac{4}{x^2})}{x^2(1+\frac{1}{x}-\frac{12}{x^2})}\\ =lim_{x\rightarrow\infty}\frac{\frac{1}{x}+\frac{4}{x^2}}{1+\frac{1}{x}-\frac{12}{x^2}}\\ =\frac{0+0}{1+0-0}=0\)

b)

horizontal asymptotes happen when the limit as x approaches infinity or negative infinity approaches some number

as found above, a horizontal asymptote is at y=0
you can also check the limit as x approaches negative infinity, but in this case it would also equal 0, so there's only one horizontal asymptote.

c)

\(\frac{x+4}{x^2+x-12}=\frac{x+4}{(x+4)(x-3)}\)

factor the numerator and denominator, if possible

you see that x+4 can be canceled out, so x=-4 must be where the hole (removeable discontinuity) is.

after canceling out the x+4 's, you're left with:
\(\frac{1}{x-3}\)

to find the coordinate of the hole, plug in x=-4 into the equation and you get y=-1/7, so, the hole is at (-4,-1/7).

 

to find the vertical asymptote, use the equation that's left over after the canceling and set the denominator equal to 0

x-3 = 0
x = 3
There is a vertical asymptote at x=3.

 Apr 21, 2020

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