How do you identify all asymptotes or holes and intercepts for #f(x)=(2x3)/(3x+1)#?
There is an asymptote at x =
1 divided by 0.10=10. Or how many 0.1s can you get in 1 1 divided by 0.01 =100.... The smaller the divisor gets, the bigger the answer
In the limit as the divisor approaches zero the result approaches infinity and that is where we have an asymptote on the graph
So as 3x+1 approaches zero the function approaches infinity
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To identify asymptotes, holes, and intercepts for ( f(x) = \frac{2x  3}{3x + 1} ):

Vertical asymptote: Set the denominator equal to zero and solve for ( x ). The vertical asymptote is at ( x = \frac{1}{3} ).

Horizontal asymptote: If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at ( y = 0 ). If the degrees are equal, divide the leading coefficients of the numerator and denominator to find the horizontal asymptote.

To find any holes, factor both the numerator and denominator, then simplify to see if there are any common factors that cancel out.

To find ( y )intercept, plug in ( x = 0 ) and solve for ( y ).

To find ( x )intercept, plug in ( y = 0 ) and solve for ( x ).
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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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