finding limits

The following methods assume the limit exists.

Method 1: Direct substitution

Direct substitution works if when the function is continuous at that point.

Method 2: Algebraic manipulation

When direct substitution results in a indeterminate form (e.g., $\frac{0}{0}$), try manipulating the polynomial (e.g., factoring and simplifying the fraction).

Warning

Always write out ${\lim }_{x\to c}$ when calculating the limit so long as there is still $x$ in the equation.

Not a polynomial 1: absolute value

When $|\cdot |$ is present in the expression, convert absolute values to their actual form in the neighborhood, ex. as $x\to 3$, $|x-8|=8-3$

Not a polynomial 2: square root in fractions, e.g. ${\lim }_{x\to 9}\frac{\sqrt{x}-3}{x-9}$

In this case, multiply by the conjugate and then continue evaluating.

Method 3: Asymptotes & Infinity

Instead of writing D.N.E. in place of asymptotes, describe the graph of the function at $x=c$. Note that the two-sided limits still have to be equal to each other for the actual limit to exist, e.g. ${\lim }_{x\to c}\frac{1}{x-1}=\mathrm{DNE}$.

Warning: They are not limits but a description of the behavior of functions,

Method 4: Graphing

Self-explanatory.

Method 5: Create a table of values as $x\to c$ from either side

For example, if $x=1$, then we can compute a table for $f(x)$ when $x$ is 0.9, 0.99, 0.999 or 1.1, 1.01, 1.001 etc.