How do you find the asymptotes for #f(x)= (2x+1)/(x1)#?
The vertical asymptote is
The horizontal asymptote is
No slant asymptote
there is no slant asymptote.
graph{(y(2x+1)/(x1))(y2)(y100x+100)=0 [18.02, 18.03, 9.01, 9.01]}
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To find the asymptotes of the function ( f(x) = \frac{2x+1}{x1} ):

Determine if there are any vertical asymptotes by identifying any values of ( x ) that make the denominator equal to zero. In this case, the denominator ( (x1) ) equals zero when ( x = 1 ). Therefore, there is a vertical asymptote at ( x = 1 ).

Determine if there are any horizontal asymptotes by examining the behavior of the function as ( x ) approaches positive or negative infinity. Use the concept of limits:
[ \lim_{x \to \infty} \frac{2x+1}{x1} ] [ \lim_{x \to \infty} \frac{2x+1}{x1} ]
To evaluate these limits, divide both the numerator and denominator by the highest power of ( x ):
[ \lim_{x \to \infty} \frac{2 + \frac{1}{x}}{1  \frac{1}{x}} ] [ \lim_{x \to \infty} \frac{2 + \frac{1}{x}}{1  \frac{1}{x}} ]
As ( x ) approaches positive or negative infinity, the terms involving ( \frac{1}{x} ) become negligible compared to other terms. Thus, the limit simplifies to:
[ \lim_{x \to \infty} \frac{2}{1} = 2 ] [ \lim_{x \to \infty} \frac{2}{1} = 2 ]
Hence, there is a horizontal asymptote at ( y = 2 ).
In summary, the function ( f(x) = \frac{2x+1}{x1} ) has a vertical asymptote at ( x = 1 ) and a horizontal asymptote at ( y = 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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