Graph the function. Be sure to plot all necessary asymptotes
H(x)=2x/x^2+x-2
The function is H(x) = 2x / (x^2 + x - 2).
First, let's find the vertical asymptotes by setting the denominator equal to zero:
x^2 + x - 2 = 0
(x + 2)(x - 1) = 0
x = -2, 1
So, the vertical asymptotes are x = -2 and x = 1.
Next, let's find the horizontal asymptote as x approaches infinity. We can do this by dividing all terms by x^2 and taking the limit as x approaches infinity:
lim(x->∞) 2x / (x^2 + x - 2)
= lim(x->∞) 2 / (1 + 1/x - 2/x^2)
= 0
Therefore, the horizontal asymptote is y = 0.
Now, plot the function by considering the asymptotes and any x-intercepts or y-intercepts. The graph should have a vertical asymptote at x = -2 and x = 1, a horizontal asymptote at y = 0, and goes through the point (2, 1) where x = 2.
The graph may look something like this:
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x = -2 x = 1
Please note that this is a rough sketch and the actual graph might differ based on the scale of the axes.