# How do you find all the asymptotes for function #y=(2x^2 + 5x- 3)/(3x+1)#?

See the explanation.

If "all the asymptotes" includes oblique asymptotes, then do the division to get:

So

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Vertical Asymptote:

Slant Asymptote:

Vertical Asymptote

To get the vertical asymptote, find the value of

The vertical asymptote is

Horizontal/Slant Asymptote

Since the degree of the numerator is greater than the degree of the denominator by one, we will get a slant asymptote. To solve this, divide

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To find all the asymptotes for the function ( y = \frac{2x^2 + 5x - 3}{3x + 1} ), follow these steps:

- Check for vertical asymptotes by setting the denominator equal to zero and solving for ( x ). Any value of ( x ) that makes the denominator zero will result in a vertical asymptote.
- Check for horizontal asymptotes by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is ( y = 0 ). If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients to find the horizontal asymptote.
- If the degrees differ by more than 1, there is an oblique (slant) asymptote. Find it by performing polynomial long division.
- Lastly, check for any asymptotes arising from holes in the graph by simplifying the function and seeing if there are any common factors between the numerator and denominator.

By following these steps, you can find all the asymptotes for the given function.

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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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