How do you find the vertical, horizontal and slant asymptotes of: #f(x)=(x^4x^2)/(x(x1)(x+2))#?
See below.
Start by simplifying the expression:
Vertical asymptotes occur where the function is undefined. This can be seen to be when
So the line
We now divide:
The quotient of this will be the oblique asymptote:
We do not need to be concerned with the remainder. So the oblique asymptote is the line:
This is the line
Graph:
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To find the vertical, horizontal, and slant asymptotes of the function ( f(x) = \frac{x^4  x^2}{x(x1)(x+2)} ), we can analyze its behavior as ( x ) approaches different values.

Vertical asymptotes: Vertical asymptotes occur where the denominator of the function becomes zero but the numerator does not. In this case, the function has vertical asymptotes at ( x = 0 ), ( x = 1 ), and ( x = 2 ) because these are the values that make the denominator zero.

Horizontal asymptote: To find the horizontal asymptote, we can look at the degrees of the numerator and denominator. Since the degree of the numerator (4) is equal to the degree of the denominator (3), there is no horizontal asymptote. Instead, there is a slant asymptote.

Slant asymptote: To find the slant asymptote, we perform polynomial division or long division to divide the numerator by the denominator. The result will be a polynomial plus a proper rational function. The polynomial part will be the equation of the slant asymptote.
Let's perform the division:
[ \frac{x^4  x^2}{x(x1)(x+2)} = x^2  2x + 4 + \frac{2x}{x1} ]
The slant asymptote is the polynomial ( y = x^2  2x + 4 ).
In summary, the function ( f(x) = \frac{x^4  x^2}{x(x1)(x+2)} ) has vertical asymptotes at ( x = 0 ), ( x = 1 ), and ( x = 2 ), no horizontal asymptote, and a slant asymptote at ( y = x^2  2x + 4 ).
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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