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

Answer 1

There exists a vertical asymptote at #x=-1/3# and a horizontal asymptote at #y=-2/3#.

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

In this case, division by zero results if the denominator of the rational function is zero, ie if #3x+1=0#, which implies if #x=-1/3#.
Horizontal asymptotes occur at #lim_(x->+_oo)g(x)=-2/3#.

The graph of the function verifies this.

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

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Answer 2

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

  1. Determine if there are any vertical asymptotes by finding the values of ( x ) that make the denominator zero.
  2. Set the denominator equal to zero and solve for ( x ).
  3. Any ( x ) values obtained in step 2 represent vertical asymptotes.
  4. Determine if there are any horizontal or slant asymptotes.
  5. To find horizontal or slant asymptotes, compare the degrees of the numerator and denominator.
  6. If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at ( y = 0 ).
  7. 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.
  8. If the degrees are equal, divide the leading coefficients to find the horizontal asymptote.

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

  1. The denominator equals zero when ( 3x + 1 = 0 ), giving ( x = -\frac{1}{3} ).
  2. Thus, there's a vertical asymptote at ( x = -\frac{1}{3} ).
  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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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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