A function and its graph are given.Use the graph to find the horizontal asymptotes,if they exist,where Confirm your results analytically.

A)
B)
C)
D)
E) no horizontal asymptotes

Question

​A function and its graph are given.Use the graph to find the horizontal asymptotes,if they exist,where  Confirm your results analytically.​ ​
​  ​
​ 
A) ​ 
B) ​ 
C) ​ 
D) ​​ 
E) ​no horizontal asymptotes

Quizplus · Accepted Answer

From the graph, we can see that as x approaches infinity or negative infinity, the function approaches the horizontal line y=1. Therefore, y=1 is the horizontal asymptote.
To confirm this analytically, let's take the limit of the function as x approaches infinity:
lim e^(-x)/ (e^(-x) + 1) as x approaches infinity
= lim 1 / ( e^(x) + 1/e^x) as x approaches infinity (dividing numerator and denominator by e^(-x))
= lim 1 / (1 + 0) as x approaches infinity (since e^x approaches infinity as x approaches infinity)
= 1
Similarly, we can show that as x approaches negative infinity, the limit of the function is also 1. Therefore, y=1 is the horizontal asymptote.

​A Function and Its Graph Are Given

A Function and Its Graph Are Given