f(x)=4x/x-2 what is the vertical asymptote I have a possible answer of x=2
To determine the vertical asymptote of a rational function, such as f(x) = 4x/(x - 2), you need to analyze the behavior of the function as x approaches certain values.
For a rational function, a vertical asymptote occurs at x = a if f(x) approaches positive or negative infinity as x approaches a from one side (left or right). Additionally, a vertical asymptote can occur at x = a if the denominator of the function becomes zero at x = a and the numerator does not.
In the given function f(x) = 4x/(x - 2), we need to find the value(s) of x that make the denominator equal to zero, i.e., solve the equation (x - 2) = 0. Solving for x, we find x = 2.
Therefore, there is a potential vertical asymptote at x = 2 since the function becomes undefined at x = 2. However, we need to further examine the limit of f(x) as x approaches 2 from the left and from the right to confirm if it approaches positive or negative infinity.
For the limit as x approaches 2 from the left:
lim[x→2^-] (4x/(x - 2))
To evaluate this limit, substitute a value slightly smaller than 2 into the function. Let's use 1.9 as an example:
lim[x→2^-] (4(1.9)/(1.9 - 2))
= (4(1.9)/(-0.1))
= -7.6/-0.1
= 76
As x approaches 2 from the left, f(x) approaches 76, not positive or negative infinity.
For the limit as x approaches 2 from the right:
lim[x→2^+] (4x/(x - 2))
To evaluate this limit, substitute a value slightly larger than 2 into the function. Let's use 2.1 as an example:
lim[x→2^+] (4(2.1)/(2.1 - 2))
= (4(2.1)/(0.1))
= 8.4/0.1
= 84
As x approaches 2 from the right, f(x) approaches 84, not positive or negative infinity.
Based on these calculations, the function f(x) = 4x/(x - 2) does not have a vertical asymptote at x = 2, as the limit does not approach positive or negative infinity.