The concept of a limit is the foundational principle upon which all of calculus is built. While algebra helps us understand a function at a specific point, limits allow us to analyze the behavior of a function near a specific point.
A limit describes the value that a function approaches as the independent variable (the input) gets closer and closer to a certain value. It is crucial to understand that the limit is not concerned with what happens at the point itself, but rather with the trend of the function as it gets arbitrarily close to that point.
Let’s investigate the behavior of the function $f$ defined by $f(x)=(x-1)/(x^2-1)$ for values of $x$ near $1$. The following table gives values of $f(x)$ for values of $x$ close to $1$ but not equal to $1$.
From the table and the graph of $f$ we see that the closer $x$ is to $1$ (on either side of $1$), the closer $f(x)$ is to $0.5$.
In fact, it appears that we can make the values of $f(x)$ as close as we like to $0.5$ by taking $x$ sufficiently close to $1$. We express this by saying “the limit of the function $f(x)=(x-1)/(x^2-1)$ as x approaches $1$ is equal to $0.5$.” The notation for this is:
$$ \lim_{x\rarr 1} \frac{(x-1)}{(x^2-1)}=0.5 $$
Suppose $f(x)$ is defined when $x$ is near the number $a$. (This means that $f$ is defined on some open interval that contains $a$, except possibly at $a$ itself.) Then we write
$$ \lim_{x\rarr a} f(x)=L $$
and say “the limit of $f(x)$, as $x$ approaches $a$, equals $L$”
This says that the values of $f(x)$ approach $L$ as $x$ approaches $a$. In other words, the values of $f(x)$ tend to get closer and closer to the number $L$ as $x$ gets closer and closer to the number $a$ (from either side of $a$) but $x \ne a$.
Also Notice the phrase “but $x$ not equal to $a$” in the definition of limit. This means that in finding the limit of $f(x)$ as $x$ approaches $a$, we never consider $x = a$. In fact, $f(x)$ need not even be defined when $x = a$. The only thing that matters is how $f$ is defined near $a$.
$f(a)= L$
$f(x)$ is not defined
$f(a)\ne L$
<aside>
Estimate the value of
$$ \lim_{t\rarr 0} \frac{\sqrt{t^2+9}-3}{t^2} $$
</aside>
<aside>
Guess the value of
$$ \lim_{x\rarr 0} \frac{\sin(x)}{x} $$
</aside>
<aside>
Find the value of
$$ \lim_{x\rarr 0} \left( x^3+ \frac{\cos(5x)}{10,000} \right) $$
</aside>