Question

Let f be the function defined by f(x)= (2^x+5)/e^x+1 for x>0. Which of the following is a horizontal asymptote to the graph of f ?

Option 1: y = 0
Option 2: y = 2/e
Option 3: y = 1
Option 4: There is no horizontal asymptote to the graph of f.

Answer 1

The horizontal asymptote to the graph of f(x) = (2^x+5)/e^x+1 is Option 4: There is no horizontal asymptote.

Step-by-step explanation:

To find the horizontal asymptote of the function f(x) = (2^x+5)/e^x+1, we need to determine the limit as x approaches positive infinity and negative infinity.

As x approaches positive infinity, both 2^x and e^x grow exponentially, so the numerator grows faster than the denominator. Therefore, the function approaches positive infinity and there is no horizontal asymptote.

As x approaches negative infinity, the exponentials decrease exponentially, so the numerator approaches 5 and the denominator approaches 1. Therefore, the function approaches 5/1 = 5. So the correct answer is Option 4: There is no horizontal asymptote to the graph of f.