Why do these laws work?

By definition of limit, if $\displaystyle\lim_{x \to a} f(x) = L$ and $\displaystyle\lim_{x \to a} g(x) = M$, then $f(x)$ is close to $L$ and $g(x)$ is close to $M$ whenever $x$ is close to $a$.

• Adding two numbers close to $L$ and $M$ gives a number close to $L+M$.


• Likewise, taking the difference of two numbers close to $L$ and $M$ gives a number close to $L-M$.


• Taking the product gives a number close to $L\cdot M$, and taking the ratio gives a number close to $L/M$ as long as $M \ne 0$.


• Multiplying $f(x)$ by a constant $c$ gives a number close to $cL$. This gives the rest of the limit laws.

When $M=0$ and/or $L=0$

The limit of a quotient rule with $M=0$ is more subtle. If $f(x)$ is close to 4 and $g(x)$ is close to 0, what can you say about $f(x)/g(x)$? It might be $3.99/(.01) = 399$, or it might by $4.01/(-.01) = -401$. Dividing by a number close to zero might give you a big positive ratio, or a big negative ratio. Without knowing more about $g(x)$, you just can't tell.

Things get even wilder when both $f(x)$ and $g(x)$ are both close to zero. In that case, $$\frac{f(x)}{g(x)} = \frac{\hbox{ small number}}{\hbox{small number}}.$$

This ratio could be $$\frac{.000001}{.001} = 0.001,$$ or it could be $$\frac{.001}{.000001} = 1000,$$ or it could be $$\frac{.001}{-0.000001} = -1000.$$ Without understanding the numerator and denominator better, we just can't tell how these "0/0" limits will behave. We'll learn more about these kinds of limits (called "indeterminate forms") in a later module. For now, we'll just file them under "do more work".