# How do you find the asymptotes for #g(x)= (-2x+3)/(3x+1)#?

There exists a vertical asymptote at

Vertical asymptotes occur at points that lead to division by zero or negative even square roots.

The graph of the function verifies this.

graph{(-2x+3)/(3x+1) [-11.25, 11.245, -5.63, 5.62]}

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To find the asymptotes for ( g(x) = \frac{-2x + 3}{3x + 1} ):

- Determine if there are any vertical asymptotes by finding the values of ( x ) that make the denominator zero.
- Set the denominator equal to zero and solve for ( x ).
- Any ( x ) values obtained in step 2 represent vertical asymptotes.
- Determine if there are any horizontal or slant asymptotes.
- To find horizontal or slant asymptotes, compare the degrees of the numerator and denominator.
- If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at ( y = 0 ).
- If the degree of the numerator is one greater than the degree of the denominator, there is a slant asymptote, which can be found by polynomial long division.
- If the degrees are equal, divide the leading coefficients to find the horizontal asymptote.

For ( g(x) = \frac{-2x + 3}{3x + 1} ):

- The denominator equals zero when ( 3x + 1 = 0 ), giving ( x = -\frac{1}{3} ).
- Thus, there's a vertical asymptote at ( x = -\frac{1}{3} ).
- The degree of the numerator is less than the degree of the denominator, so there's a horizontal asymptote at ( y = 0 ).

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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.

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- How do you find the vertical, horizontal and slant asymptotes of: #y=(2x )/ (x-5)#?
- How do you describe the transformation in #y=1/3x^3+2#?

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