quick summary of limits

A limit is the value a function approaches as its input approaches a certain value. The notation is as follows:

$$ \lim_{x\to c}f(x) = L $$

This means that the limit of $f(x)$ as $x$ approaches $c$ is $L$. For example, $\displaystyle\lim_{x\to 2}(2x) = 4$ because as $x$ approaches 2, $2x$ approaches 4.

basic methods of solving limits

Direct substitution: If $f(x)$ is continuous at $x = c$, then $\displaystyle\lim_{x\to c}f(x) = f(c)$.

• Example: $\displaystyle\lim_{x\to 1}(x^2 + 2x + 3)$

Factoring: You can sometimes use factoring to simplify $f(x)$ into an equivalent expression, then use direct substitution.

• Example: $\displaystyle\lim_{x\to 1}\frac{x^2-1}{x+1}$

Multiplying by conjugates

• Example: $\displaystyle\lim_{x\to 4}\frac{\sqrt{x-3}-1}{x-4}$

Trig identities

• Example: $\displaystyle\lim_{x\to\pi}\frac{\sin(2x)}{\sin(x)}$
• Limit properties

the $\frac{\sin(x)}{x}$ limit

This limit is very important in calculus:

$$ \lim_{x\to 0}\frac{\sin(x)}{x} = 1 $$

This limit allows us to find many limits involving sines.

• Example: $\displaystyle\lim_{x\to 0}\frac{\sin(x^3)}{\sin(x^2)}$

other important limits