How do you find the asymptotes for #(e^x)/(1+e^x)#?
There is no vertical asymptote. (assuming we are restricted to the Real number plane)
Horizontal asymptotes at
For verification purposes, here's what the graph looks like: graph{e^x/(1+e^x) [6.24, 6.244, 3.12, 3.12]}
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To find the asymptotes of the function ( \frac{e^x}{1+e^x} ), we need to examine its behavior as ( x ) approaches positive or negative infinity.

As ( x ) approaches positive infinity: [ \lim_{x \to \infty} \frac{e^x}{1+e^x} = \lim_{x \to \infty} \frac{e^x}{e^x(1+e^{x})} = \lim_{x \to \infty} \frac{1}{1+e^{x}} = 1 ] So, there is a horizontal asymptote at ( y = 1 ).

As ( x ) approaches negative infinity: [ \lim_{x \to \infty} \frac{e^x}{1+e^x} = \lim_{x \to \infty} \frac{1}{e^{x}+1} = 0 ] So, there is a horizontal asymptote at ( y = 0 ).
Therefore, the horizontal asymptotes of ( \frac{e^x}{1+e^x} ) are ( y = 1 ) and ( y = 0 ).
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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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