How do you find the vertical, horizontal or slant asymptotes for #(2x+sqrt(4x^2)) / (3x^2x2)#?
Note that the curve does not reach the xaxis for positive intercept.
Important note:
troubled me in making my data compatible with that in the graph. I
had to take negative root, breaking the convention.
graph{y(x1)(3x2)2xsqrt(4x^2)=0 [10.17, 10.17, 5.105, 5.065]}
graph{y(x1)(3x2)2xsqrt(4x^2)=0 [20.34, 20.34, 10.19, 10.15]}
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To find the vertical, horizontal, or slant asymptotes of the function ( \frac{2x + \sqrt{4  x^2}}{3x^2  x  2} ), we'll analyze the behavior of the function as ( x ) approaches different values.

Vertical Asymptotes: Vertical asymptotes occur where the denominator of the function becomes zero, but the numerator does not. So, we set the denominator equal to zero and solve for ( x ): [ 3x^2  x  2 = 0 ] Solving this quadratic equation, we find the roots: [ x = \frac{1 \pm \sqrt{1 + 24}}{6} ] [ x = \frac{1 \pm \sqrt{25}}{6} ] [ x = \frac{1 \pm 5}{6} ] [ x = \frac{2}{3}, , \frac{1}{3} ] So, the vertical asymptotes are at ( x = \frac{2}{3} ) and ( x = \frac{1}{3} ).

Horizontal Asymptote: To find the horizontal asymptote, we examine the behavior of the function as ( x ) approaches positive or negative infinity. We'll perform long division or divide the leading terms of the numerator and denominator: [ \lim_{x \to \pm \infty} \frac{2x + \sqrt{4  x^2}}{3x^2  x  2} ] Since the degree of the numerator is 1 and the degree of the denominator is 2, the horizontal asymptote is at ( y = 0 ).

Slant Asymptote: To find the slant asymptote, we perform polynomial long division or use synthetic division if the degree of the numerator is one greater than the degree of the denominator. If the degree of the numerator is two or more greater, then no slant asymptote exists.
In this case, since the degree of the numerator is less than or equal to the degree of the denominator, there is no slant asymptote.
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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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