How do you find the asymptotes for #g(x)=x/(root4(x^4+2))#?
Evaluate the limits of
graph{x/root(4)(x^4+2) [5.55, 5.55, 2.775, 2.774]}
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To find the asymptotes of the function (g(x) = \frac{x}{\sqrt[4]{x^4 + 2}}), you need to consider both horizontal and vertical asymptotes.

Vertical Asymptotes: Vertical asymptotes occur where the denominator becomes zero. However, in this case, the denominator is a fourth root, and the expression inside the root, (x^4 + 2), is always positive for real (x) because any real number raised to an even power is nonnegative, and adding 2 to it ensures it's always positive. Therefore, there are no vertical asymptotes for this function.

Horizontal Asymptotes: To find horizontal asymptotes, analyze the behavior of the function as (x) approaches positive or negative infinity. As (x) approaches infinity, (x) dominates the expression. Since both the numerator and the denominator approach infinity, the function has a horizontal asymptote at (y = 1), as the term inside the root becomes insignificant compared to (x).
Therefore, the horizontal asymptote of the function (g(x)) is (y = 1), and there are no vertical asymptotes.
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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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