Understanding Vertical Asymptotes at x=5 in Rational Functions: A Guide.

Vertical Asymptote at x = 5

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A vertical asymptote at x = 5 indicates that the function has decreasing or increasing values as we approach x = 5, but the function shoots to positive or negative infinity at x = 5 and cannot be evaluated at x = 5. In other words, the graph of the function has a vertical line at x = 5 which the function cannot cross or touch.

One common type of function that has a vertical asymptote at x = 5 is a rational function (a function with a polynomial in the numerator and denominator). To identify if a rational function has a vertical asymptote at x = 5, we need to check if there is a factor of (x – 5) in the denominator that is not cancelled out by a matching factor in the numerator (i.e., cannot be factored out), because this causes the denominator to be zero when x = 5.

For example, the function f(x) = (x^2 + 2x – 15) / (x – 5) has a vertical asymptote at x = 5 because when x = 5, the denominator becomes zero and the function blows up. On the other hand, the function g(x) = (x^2 – 25) / (x – 5) does not have a vertical asymptote at x = 5 because the factor (x – 5) in the denominator can be cancelled out by the factor (x – 5) in the numerator, leaving no zero in the denominator.

It is important to note that not all functions with a vertical line at x = 5 have a vertical asymptote. For example, the function h(x) = sin(x – 5) + 2 has a vertical line at x = 5 (because the period of sin(x – 5) is 2π and it crosses the x-axis at x = 5), but it does not have a vertical asymptote at x = 5 because the function is bounded and continuous at x = 5.

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