Consider the function $f(x)=\frac{1}{1+a e^{-x}}$ , for $a0$ .As a increases, what happens to the horizontal asmyptotes of f?
A)They shift upward.
B)They remain the same.
C)They shift downward.

Question

Consider the function $f(x)=\frac{1}{1+a e^{-x}}$ , for $a>0$ .As a increases, what happens to the horizontal asmyptotes of f?
A)They shift upward.
B)They remain the same.
C)They shift downward.

Quizplus · Accepted Answer

The horizontal asymptotes of the function $f(x)=\frac{1}{1+a e^{-x}}$ are determined by the behavior of the function as $x$ approaches positive and negative infinity. As $x$ approaches positive infinity, $e^{-x}$ approaches 0, making the function approach $\frac{1}{1+0} = 1$, regardless of the value of $a$. As $x$ approaches negative infinity, $e^{-x}$ grows without bound, but since it's in the denominator, the function's value approaches 0. Therefore, the horizontal asymptotes are $y=1$ and $y=0$, which do not depend on the value of $a$, indicating that the horizontal asymptotes remain the same as $a$ increases.

Consider the Function f(x)=11+ae−xf(x)=\frac{1}{1+a e^{-x}}f(x)=1+ae−x1​ , For a>0a>0a>0 As a Increases, What Happens to the Horizontal Asmyptotes of Increases

Consider the Function $f(x)=\frac{1}{1+a e^{-x}}$ , For $a>0$ As a Increases, What Happens to the Horizontal Asmyptotes of Increases