How to find the asymptotes of #f(x) = (x+6)/(2x+1)# ?
This function has vertical asymptote
To check if a rational function has a vertical asymtote(s) you have to look for zeroes of the denominator.
To look for the horizontal asymptotes you have to calculate
In this example we have:
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To find the asymptotes of ( f(x) = \frac{x+6}{2x+1} ), we examine the behavior of the function as ( x ) approaches positive and negative infinity.

Vertical Asymptote: Set the denominator equal to zero and solve for ( x ). Any ( x ) value that makes the denominator zero will create a vertical asymptote. [ 2x + 1 = 0 ] [ x = \frac{1}{2} ] Thus, there is a vertical asymptote at ( x = \frac{1}{2} ).

Horizontal Asymptote: To find the horizontal asymptote, we look at 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 at ( y = 0 ). If the degree of the numerator is equal to the degree of the denominator, divide the leading coefficients. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. In this case, the degree of the numerator and denominator are both 1. So, we divide the leading coefficients: [ \lim_{x \to \pm \infty} \frac{x + 6}{2x + 1} = \frac{1}{2} ] Therefore, the horizontal asymptote is ( y = \frac{1}{2} ).
Hence, the asymptotes of ( f(x) = \frac{x+6}{2x+1} ) are:
 Vertical asymptote: ( x = \frac{1}{2} )
 Horizontal asymptote: ( y = \frac{1}{2} )
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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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