Evaluating Indeterminate Limits

Evaluating Indeterminate Limits

A function doesn't have to go to $\pm\infty$ at a discontinuity though. Consider, for instance, evaluating $\lim\limits_{x\to3}\dfrac{x^2-x-6}{x-3}$ Plugging in $x=3$ gives us “$0/0$”, which is known as an indeterminate form.

A good first step is to factor where possible. Once we cancel the factor $x-3$ on top and bottom, this function is just the function $f(x)=x+2$, but with a hole at $x=3$ The hole is there because we cannot ignore that the original form of the function had issues at $x=3$

The expressions agree for $x\ne3$, which is enough to find the limit as $x$ approaches 3. Cancelling a factor does not define the original function at 3.

For a square-root numerator, try multiplying by its conjugate. For a fraction within a fraction, first combine the numerator over a common denominator. The practice below works through all three methods.