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

Answer 1

See the explanation.

The line #x=-1/3# is a vertical asymptote because the denominator, but not the numerator is #0# at #-1/3#
There is no horizontal asymptote, because as #x# increases [decreases] without bound, #y# also increases [decreases] without bound.

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

#(2x^2+5x-3)/(3x+1) = 2/3x+13/9-(40/9)/(3x+1)#

So

#y=2/3x+13/9# is an oblique asymptote.
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Answer 2

Vertical Asymptote: #x=-1/3#
Slant Asymptote: #y=2/3x+13/9#.

Vertical Asymptote
To get the vertical asymptote, find the value of #x# that will make the denominator #0# (undefined).

#3x+1=0#
#3x=-1#
#x=-1/3#

The vertical asymptote is #color(blue)(x=-1/3)#.

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 #2x^2+5x-3# by #3x+1# using long division. The answer will be your slant asymptote.

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Answer 3

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

  1. 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.
  2. 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.
  3. If the degrees differ by more than 1, there is an oblique (slant) asymptote. Find it by performing polynomial long division.
  4. 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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Answer from HIX Tutor

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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