Infinite Limits

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Infinite Limits

Infinite limits exist when we can plug in a number for x that causes the denominator of a rational function in lowest terms to equal zero.

Here is an example of a rational function in lowest terms, which means that we cannot factor and cancel anything in the fraction.

We can see that setting x=1 gives zero in the denominator, which means that we have a vertical asymptote at x=1, and therefore an infinite limit at that point.

Now that we’ve established that this is a rational function in lowest terms and that a vertical asymptote exists, all that’s left to determine is whether the limit at x=1 approaches positive or negative infinity.

In order to do that, simply plug in a number very close to one. If our result is positive, the limit will be positive infinity. If the result is negative, the limit is negative infinity.

We can see that the result will be very large and positive, so we know that the limit of this function at x=1 is positive infinity.