Limit Laws and Computations

Intuitive idea of why 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 $a$.

The limit laws not involving a quotient:

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 $cL$.

The limit laws involving a quotients:

Taking the quotient of two numbers close to $L$ and $M$ gives a number close to $\frac{L}{M}$ as long as $M$ is not zero.

The quotient $\frac{f(x)}{g(x)}$ is more subtle when the denominator approaches 0, or both the numerator and denominator approach $0$, so that you cannot divide:

What happens when the limit looks like $\frac{\text{non-zero number}}{0}$, because $g$ is approaching zero but $f$ is not? Consider the example where $f(x)$ is close to 4 and $g(x)$ is close to 0. It might be $\frac{3.99}{.000001} = 3990000$, or it might be $\frac{4.01}{-.000001} = -4010000$. Dividing by a number close to zero might give you a big positive ratio, or a big negative ratio. Without knowing more about $f(x)$ and $g(x)$, you just can't tell. What we do know is that if the denominator goes to zero and the numerator does not, we have a vertical asymptote, and the limit must be either $\infty$, $-\infty$ or it does not exist. So $\frac{\text{non-zero number}}{0}$ is not an indeterminate form -- we know this limit value will be one of the three values indicated above. To determine which of the three is the value of the limit, we need only compute the left- and right-hand limits.

What happens when the limit looks like $\frac{0}{0}$, because both $f$ and $g$ both approach zero and either could be positive or negative? 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 limits of the form $\frac{0}{0}$ will behave. We'll learn more about these kinds of limits; $\frac{0}{0}$ is an indeterminate form, which means that we cannot determine its value, or even make a good guess, without more effort. To evaluate indeterminate forms, we will do more work (which means do algebra to change the form of $\frac{f(x)}{g(x)}$.