How do you find vertical, horizontal and oblique asymptotes for #f(x) = (3x+5)/(x-6)#?
vertical asymptote at x = 6
Equating the denominator to zero and solving for x yields the value that x cannot be; if the numerator is non-zero for this value, then it is a vertical asymptote. The denominator of f(x) cannot be zero because doing so would render f(x) undefined.
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To find the vertical asymptotes of the function (f(x) = \frac{3x + 5}{x - 6}), we look for values of (x) that make the denominator (x - 6) equal to zero. So, we set (x - 6 = 0) and solve for (x). This gives us (x = 6). Therefore, the vertical asymptote of the function is (x = 6).
To find horizontal and oblique asymptotes, we examine the behavior of the function as (x) approaches positive and negative infinity.
As (x) approaches positive or negative infinity, the term with the highest power in the numerator and the denominator dominates the function. In this case, the term with the highest power in the numerator and the denominator is (x).
Therefore, we divide both the numerator and the denominator by (x), and then we examine the limit as (x) approaches infinity:
(f(x) = \frac{3x + 5}{x - 6} = \frac{3 + \frac{5}{x}}{1 - \frac{6}{x}})
As (x) approaches infinity, (\frac{5}{x}) and (\frac{6}{x}) approach zero, and we're left with:
(f(x) \approx \frac{3}{1})
So, the horizontal asymptote is (y = 3).
Since the degree of the numerator is one more than the degree of the denominator, we have an oblique asymptote. To find the equation of the oblique asymptote, we perform polynomial long division or synthetic division. After dividing (3x + 5) by (x - 6), we get:
(3x + 5 = (x - 6)(3) + 23)
So, the oblique asymptote is (y = 3x + 23).
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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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