Vertical asymptote at x = a
Function undefined at x = a and lim as x →a = ±∞ on either side of a
When a function has a vertical asymptote at x = a, it means that the function approaches either positive or negative infinity as x approaches a from both sides (either the left or the right). A vertical asymptote is a vertical line that the function gets closer and closer to but never touches.
The reason for a vertical asymptote can be that the denominator of a rational function becomes zero at x = a, causing the whole function to become undefined. For example, consider the rational function f(x) = (x^2-4)/(x-2). This function has a vertical asymptote at x = 2, because when x gets very close to 2 from both the left and the right sides, the denominator becomes very close to zero, making the whole function get very large.
It is important to note that not all functions have vertical asymptotes. For example, polynomials, square roots, and exponentials do not have vertical asymptotes.
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