We can note that the smaller the distance of the input value $x$ to $2$, the smaller the distance of the output to $9$. This is most commonly described in the terms "As $x$ approaches $2$, $f(x)$ approaches $9$", or "As $x \to 2$, $f(x) \to 9$."
Limits are valuable because they can be used to describe a point on a graph, even if that point is not present.
If they have a limit of the form $lim_{x \to a} \frac{f(x)}{g(x)}$ where both $f(x) \to 0$ and $g(x) \to 0$ as $x \to a$ then this limit **may or may not** exist and is said to be an indeterminate form of type $\dfrac{0}{0}$.
To find this limit if it exists we must perform some mathematical manipulations on the quotient in order to change the form of the function. Some of the manipulations that can be tried are:
- Factor or Foil polynomials and try dividing out a common factor.
- Multiply numerator and denominator by the conjugate of a radical expression
- Combine fractions in the numerator or denominator of a complex fraction
If we have a limit of the form $\lim_{x \to a} \frac{f(x)}{g(x)}$ where both $f(x) \to \infty$ and $g(x) \to \infty$ as $x \to a$ then the limit may or may not exist and is said to be an indeterminate form of type $\frac{\infty}{\infty}$.
To find the limit if it exists we must perform some algebraic manipulations on the quotient in order to change the form of the function.
If $f(x)$ and $g(x)$ are polynomials, then we can multiply the numerator and denominator by $\dfrac{1}{x^n}$, where $n$ is the degree of the polynomial in the denominator.
Let $f$ be a continuous function on the interval $[a, b]$ and let $N$ be any number strictly between $f(a)$ and $f(b)$. Then there exists a number $c$ in $(a, b)$ such that $f(c) = N$.
