the graph has a vertical asymptote at x=a
lim x->a = +- ∞
A vertical asymptote at x = a means that the function is approaching infinity or negative infinity as we approach the value of x = a.
One reason why a vertical asymptote might occur is if the denominator of a rational function, such as f(x) = g(x)/h(x), becomes zero at x = a. In this case, as we approach x = a from both sides, the absolute value of the denominator gets smaller and smaller, causing the value of the function to either approach positive or negative infinity.
Another reason for a vertical asymptote is if the function has a nonremovable discontinuity at x = a. This can happen if the limit of the function doesn’t exist at x = a or if the function has a jump discontinuity at that point.
To determine the nature of a vertical asymptote at x = a, we can evaluate the limits of the function as x approaches a from both sides. If both of the limits approach infinity or negative infinity, then we have a vertical asymptote. If one or both of the limits are finite, then we do not have a vertical asymptote at x = a.
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