Understanding Asymptotes in Rational Functions: Why ( f(x) frac{3x^2 14x - 5}{x - 5} ) Does Not Have a Vertical Asymptote at ( x -5 )

When dealing with rational functions, one of the key concepts to understand is the presence of vertical asymptotes. A vertical asymptote occurs when the function approaches infinity or negative infinity as ( x ) approaches a certain value. This happens typically when the denominator of a rational function equals zero, making the function undefined at that specific point. In this article, we'll explore why the function ( f(x) frac{3x^2 14x - 5}{x - 5} ) does not have a vertical asymptote at ( x -5 ).

Factoring the Function

To properly analyze the function, it is important to correctly format and factor it. The original function is given as:

Correct Form: ( f(x) frac{3x^2 14x - 5}{x - 5} )

Starting by factoring the numerator, we find:

( 3x^2 14x - 5 (x 5)(3x - 1) )

Thus, the function can be written as:

Factored Form: ( f(x) frac{(x 5)(3x - 1)}{x - 5} )

Analysis of the Factored Function

When analyzing the factored function, it becomes clear that the original function ( f(x) ) has a singularity at ( x 5 ) due to the ( x - 5 ) in the denominator. However, at ( x -5 ), both the numerator and the denominator become zero, simplifying the function to:

At ( x -5 ): ( f(x) frac{(x 5)(3x - 1)}{x - 5} ) becomes ( frac{0 cdot (3(-5) - 1)}{-10} 0 )

This indicates that ( x -5 ) is not a vertical asymptote but a point where the function simplifies to another form.

Limit Analysis

To further confirm, we can analyze the limit of the function as ( x ) approaches (-5). The limit does not approach infinity or negative infinity, but rather a finite value. Therefore, the function does not have a vertical asymptote at ( x -5 ).

The limit can be calculated as:

Calculation: ( limlimits_{x to -5} frac{(x 5)(3x - 1)}{x - 5} 3(-5) - 1 -16 )

This finite limit implies that the function does not blow up or go to infinity as ( x ) approaches (-5).

Conclusion

Given the analysis above, we can conclude that the function ( f(x) frac{3x^2 14x - 5}{x - 5} ) does not have a vertical asymptote at ( x -5 ). Instead, it simplifies to ( f(x) 3x - 1 ) for all ( x eq 5 ), and it has a singular point at ( x 5 ) due to the denominator being zero.

This example highlights the importance of proper function formatting and the distinction between singular points and vertical asymptotes in rational functions.

Key Learning Points

Singular Point vs. Vertical Asymptote: A singular point occurs when the function is undefined due to a zero in the denominator, but the function approaches a finite value. A vertical asymptote occurs when the function approaches infinity or negative infinity. Factoring Numerators: Properly factoring the numerator can simplify the function and indicate where it simplifies to another form rather than approaching infinity. Limits: Analyzing the limit of the function as ( x ) approaches a specific value can determine whether a function has a vertical asymptote or not.

Keywords: vertical asymptote, rational function, limits, asymptote analysis