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We give basic laws for working with limits.

In the previous section were able to compute the limits

using continuity of the functions $x^3$, $\sqrt {2}$, and $\cos {x}$, at $x=\pi$. Does this imply that we can compute the limits

Well, we cannot use continuity here, because we don’t know if the functions $x^3-\cos {x}$, $\frac {\sqrt {2}}{\cos {x}}$, $\sqrt {2}\cdot x^3\cdot \cos {x}$, and $\cos ({x^3})$ are continuous at $x=\pi$, and

we have no other tools available, since the graphs and tables are not reliable. Obviously, we need more tools to help us with computation of limits.

In this section, we present a handful of rules, called the Limit Laws, that allow us to find limits of various combinations of functions.

In plain language, “limit of a sum equals the sum of the limits,” “limit of a product equals the product of the limits,” etc.

Let’s examine how the Limit Laws can be used in computation of limits.

We can generalize the example above to get the following theorems.

Where is $f(x) = \frac {x^2-3x+2}{x-2}$ continuous?
for all real numbers at $x=2$ for all real numbers, except $x=2$ impossible to say
True or false: If $f$ and $g$ are continuous functions on an interval $I$, then $f\pm g$ is continuous on $I$.
True False
True or false: If $f$ and $g$ are continuous functions on an interval $I$, then $f/g$ is continuous on $I$.
True False

We still don’t know how to compute a limit of a composition of two functions. Our next theorem provides basic rules for how limits interact with composition of functions.

Because the limit of a continuous function is the same as the function value, we can now pass limits inside continuous functions.

Using the Composition Limit Law, we can compute the last example from the beginning of this section.

Many of the Limit Laws and theorems about continuity in this section might seem like they should be obvious. You may be wondering why we spent an entire section on these theorems. The answer is that these theorems will tell you exactly when it is easy to find the value of a limit, and exactly what to do in those cases.

The most important thing to learn from this section is whether the limit laws can be applied for a certain problem, and when we need to do something more interesting. We will begin discussing those more interesting cases in the next section.

### A list of questions

Let’s try this out.

Can this limit be directly computed by limit laws?
yes no
Compute:
Can this limit be directly computed by limit laws?
yes no
Can this limit be directly computed by limit laws?
yes no
Can this limit be directly computed by limit laws?
yes no
Can this limit be directly computed by limit laws?
yes no
Compute:
Can this limit be directly computed by limit laws?
yes no
Can this limit be directly computed by limit laws?
yes no
Can this limit be directly computed by limit laws?
yes no
Compute:
Can this limit be directly computed by limit laws?
yes no