How do you find all the asymptotes for function #f(x) = (2x^2+3x+8)/(x+3)#?
Vertical asymptote
There are three types of asymptotes.
It is apparent that we have this only when degree of numerator is one more than that of denominator.
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To find the asymptotes of the function ( f(x) = \frac{2x^2+3x+8}{x+3} ), we need to consider both vertical and horizontal asymptotes.
Vertical Asymptotes: Vertical asymptotes occur where the denominator of the function becomes zero but the numerator doesn't. For this function, the vertical asymptote occurs when ( x + 3 = 0 ), so ( x = -3 ) is the vertical asymptote.
Horizontal Asymptote: To find the horizontal asymptote, we need to examine the behavior of the function as ( x ) approaches positive or negative infinity. We can use polynomial division or limits to determine this.
When ( x ) approaches positive or negative infinity, the highest power terms dominate the function. In this case, as ( x ) goes to infinity, ( 2x^2 ) dominates the function. So, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator, which is ( y = \frac{2}{1} = 2 ).
Therefore, the asymptotes for the function ( f(x) = \frac{2x^2+3x+8}{x+3} ) are:
- Vertical asymptote: ( x = -3 )
- Horizontal asymptote: ( y = 2 )
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When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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