Answer:

1.(x^2 - 2)/(2x - 4) =>

2(x - 2) = 0

x = 2 => first we know there is a vertical asymptote at x = 2

Now since the degree of the numerator is greater than the degree of the denominator we divide to get:

(x^2 - 2) / (2x-4) = x/2 + 1 + 1/(x - 2)

The result above is the exact expression we started with just written differently, the quotient part is the line:

y = (1/2)x + 1 => slope = 1/2, x-intercept is at: (-2, 0), y-intercept is at(0, 1)

It is also the slant/oblique asymptote we conclude that the above graph is asymptotic to the line:

y = (1/2)x + 1

Please see the graph in the link below:

plot{y = (x^2-2) / (2x-4), y = (1/2)x + 1} -

The vertical asymptote is when the denominator is zero at x = 2

For oblique or horizontal asymptotes you should do a long division.

2.)

f(x)=122x2–42x−4f(x)=122x2–42x−4

=12x(2x−4)+2(2x−4)+42x−4=12x(2x−4)+2(2x−4)+42x−4

=12(x+2)+22x−4,=12(x+2)+22x−4,

 the oblique asymptote is y=12(x+2)

Step-by-step explanation:

Let me assume you wrote the question incorrectly, but you meant to put the 4 in the denominator. Then you MUST put a parentheses like this:

y = 1/(x+4)

If that is what happened here, and you wanted the entire expression (x+4) in the denominator, then if x=-4 the expression is undefined (there is no such thing as “divide by zero”, it is undefined). x=-4 will then be the vertical line where the function is undefined. That is what we would call a “vertical asymptote”.

Keep in mind, if that is what you meant, you wrote it incorrectly. Don’t make excuses; edit the question and put (x+4) in parentheses.

If it was written using math typeset (LaTeX) you would not need the parentheses because it would look like this:

When written this way, it is clear that the entire factor, x+4, is in the denominator.

Maybe you actually did not make a mistake in how you wrote it. Maybe there is no parentheses needed. In that case we are dividing 1 by x, and then adding just 4 to that answer. In math typeset (LaTeX), it looks like this:

In that case 1/x is undefined if x=0. The vertical asymptote is at x=0. In that case, if we leave out the extra parentheses, we would have a completely different function and a completely different answer

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