Understanding Horizontal Asymptotes of the Function ( f(x) frac{100}{2.5 - e^{-3x}} )

In this article, we will explore how to find the horizontal asymptotes of the function ( f(x) frac{100}{2.5 - e^{-3x}} ) by analyzing the behavior of the function as ( x ) approaches positive and negative infinity. This process involves taking limits and understanding the behavior of exponential functions.

Step 1: Analyzing as ( x rightarrow infty )

When ( x ) approaches positive infinity, the term ( e^{-3x} ) approaches 0 because the exponent ( -3x ) becomes a very large negative number, causing the exponential term to shrink to nearly zero.

As a result, the denominator ( 2.5 - e^{-3x} ) approaches 2.5. Substituting this into the function gives:

[ f(x) rightarrow frac{100}{2.5} 40 ]

Step 2: Analyzing as ( x rightarrow -infty )

When ( x ) approaches negative infinity, the term ( e^{-3x} ) grows very large because the exponent ( -3x ) becomes a very large positive number, causing the exponential term to grow exponentially large. This results in:

[ 2.5 - e^{-3x} rightarrow -infty ]

Substituting this into the function gives:

[ f(x) rightarrow frac{100}{-infty} 0 ]

Conclusion

The horizontal asymptotes of the function ( f(x) frac{100}{2.5 - e^{-3x}} ) are determined as follows:

( y 40 ) as ( x rightarrow infty ) ( y 0 ) as ( x rightarrow -infty )

Therefore, the horizontal asymptotes are ( y 40 ) and ( y 0 ).

Additional Inspections

We can further analyze the function by rewriting it in different forms to get a clearer picture of its behavior.

Starting with the function:

[ f(x) frac{100}{2.5 - e^{-3x}} frac{100}{2.5 - frac{1}{e^{3x}}} ]

When ( x ) is a large negative number, the denominator becomes very large, causing the entire fraction to approach 0. This confirms that the horizontal axis ( y 0 ) is a horizontal asymptote:

[ f(x) rightarrow 0 ]

When ( x ) is a large positive number, the term ( e^{-3x} ) in the denominator approaches 0, making the denominator ( 2.5 - e^{-3x} ) approach ( 2.5 ), and thus:

[ f(x) rightarrow frac{100}{2.5} 40 ]

This confirms that ( y 40 ) is a horizontal asymptote.

Plotting the Function and Asymptotes

To better visualize the function and its asymptotes, we can plot the function ( f(x) frac{100}{2.5 - e^{-3x}} ). Additionally, the horizontal asymptotes ( y 0 ) and ( y 40 ) can be seen as dashed lines.

To find a vertical asymptote, we set the denominator equal to zero:

[ 2.5 - e^{-3x} 0 ]

Solving for ( x ), we get:

[ e^{-3x} 2.5 ]

[ -3x ln{2.5} ]

[ x -frac{1}{3} ln{2.5} approx -0.41759 ]

The vertical asymptote is thus located at ( x approx -0.41759 ).