Limits

Let $f$ be a function such that $f(x)$ is defined for $x$ arbitrarily close to but different from $a.$ Then, if $f(x)$ gets arbitrarily close to the number $L$ for $x$ sufficiently close to (but not equal to) $a$ regardless from which side, we say that $f(x)$ approaches $\bold{L}$ as $\bold{x \to a}$ ("$x$ approaches $a$") or that the limit of $f(x)$ as $x \to a$ is $L$.

In other words, we can make $f(x)$ be as close to $L$ as we like by choosing any $x$ in the domain of $f$ sufficiently close to but not equal to $a.$ We write

$\displaystyle \lim_{x \to a} f(x) = L$

or

$f(x) \to L$ as $x \to a.$        The limit of $f(x)$ as $x \to a$ is $L$.

If $f(x)$ fails to approach a single fixed number as $x$ approaches $a$, then we say that $f(x)$ has no limit as $x \to a$, or

$\displaystyle \lim_{x \to a} f(x)\ $ does not exist.

Estimating limits numerically

To numerically estimate $\displaystyle \lim_{x \to a} f(x)$ if it exists:

• Make a table of values for $f(x)$ using values of $x$ that approach $a$ closely from either side.
• If the limit exists, then the values of $f(x)$ will approach the limit as $x$ approaches $a$ from both sides.
• The more accurately you wish to evaluate this limit, the closer to $a$ you will need to choose the values of $x$.

One-sided limits

Let $f$ be a function such that $f(x)$ is defined for $x$ arbitrarily close to and on the left of $a.$ Then, if $f(x)$ gets arbitrarily close to the number $L$ for $x$ sufficiently close and on the left of $a$, we say that $f(x)$ approaches $\bold{L}$ as $\bold{x \to a^-}$ ("$x$ approaches $a$ from the left") or that the limit of $f(x)$ as $x \to a^-$ is $L$.

In other words, we can make $f(x)$ be as close to $L$ as we like by choosing any $x$ in the domain of $f$ sufficiently close to and to the left of $a.$ We write

$\displaystyle \lim_{x \to a^-} f(x) = L$

or

$f(x) \to L$ as $x \to a^-.$
The limit of $f(x)$ as $x \to a$ from the left is $L$.

Similarly, if $f(x)$ is defined for $x$ arbitrarily close to and on the right of $a$ and $f(x)$ gets arbitrarily close to $L$ for $x$ sufficiently close and on the right of $a$, we say that $f(x)$ approaches $\bold{L}$ as $\bold{x \to a^+}$ ("$x$ approaches $a$ from the right") or that the limit of $f(x)$ as $x \to a^+$ is $L$. In other words, we can make $f(x)$ be as close to $L$ as we like by choosing any $x$ in the domain of $f$ sufficiently close to and to the right of $a,$ and we write

$\displaystyle \lim_{x \to a^+} f(x) = L$

or

$f(x) \to L$ as $x \to a^+.$
The limit of $f(x)$ as $x \to a$ from the right is $L$.

If $f(x)$ approaches the same limit $L$ as $a \to a^-$ and as $x \to a^+$, then $\displaystyle \lim_{x \to a}f(x) = L$ as well:

$\displaystyle \lim_{x \to a}f(x) = \lim_{x \to a^-}f(x) = \lim_{x \to a^+}f(x) = L.$

If $f(x)$ approaches different limits as $a \to a^-$ and as $x \to a^+$, then $\displaystyle \lim_{x \to a}f(x)$ does not exist.

Limits at infinity

Let $f$ be a function such that $f(x)$ is defined for $x$ arbitrarily large and positive. Then, if $f(x)$ gets arbitrarily close to the number $L$ for $x$ sufficiently large and positive, we say that $f(x)$ approaches $\bold{L}$ as $\bold{x \to +\infty}$ or that the limit of $f(x)$ as $x \to +\infty$ is $L$.

In other words, we can make $f(x)$ be as close to $L$ as we like by choosing any $x$ in the domain of $f$ sufficiently large and positive. We write

$\displaystyle \lim_{x \to +\infty} f(x) = L$

or

$f(x) \to L$ as $x \to +\infty.$
The limit of $f(x)$ as $x \to +\infty$ is $L$.

Similarly, if $f(x)$ is defined for $x$ arbitrarily large numerically but negative, and $f(x)$ gets arbitrarily close to $L$ for $x$ sufficiently large numericaly and negative, we say that $f(x)$ approaches $\bold{L}$ as $\bold{x \to -\infty}$ or that the limit of $f(x)$ as $x \to -\infty$ is $L$. In other words, we can make $f(x)$ be as close to $L$ as we like by choosing any $x$ in the domain of $f$ sufficiently large numerically and negative, and we write

$\displaystyle \lim_{x \to -\infty} f(x) = L$

or

$f(x) \to L$ as $x \to -\infty.$
The limit of $f(x)$ as $x \to -\infty$ is $L$.

Estimating limits graphically

To decide whether $\displaystyle \lim_{x \to a}f(x)$ exists and to estimate its value if it does:

1. Draw the graph of $f(x)$ by hand or with graphing technology.
2. Position your pencil point (or the trace cursor) on a point of the graph to the left of $x = a$.
3. Move the point along the graph toward $x = a$ from the left, and read the $y$-coordinate as you go. The value that the $y$-coordinate approaches (if any) is the limit $\displaystyle \lim_{x \to a^{-}}f(x)$.
4. Repeat Steps 2 and 3, this time starting from a point on the graph to the right of $x = a$ and approaching $x = a$ along the graph from the right. The value that the $y$-coordinate approaches (if any) is $\displaystyle \lim_{x \to a^+}f(x).$
5. If the left and right limits both exist and have the same value $L$, then $\displaystyle \lim_{x \to a}f(x) = L.$ If their values differ, the limit does not exist. The value $f(a)$ has no relevance whatsoever.
6. To evaluate $\displaystyle \lim_{x \to +\infty}f(x)$, move the pencil point toward the far right of the graph, and estimate the value the $y$-coordinate approaches (if any). For $\displaystyle \lim_{x \to -\infty}f(x)$, move the pencil point toward the far left.

Computing limits as x → a algebraically

To decide algebraically whether $\displaystyle \lim_{x \to a}f(x)$ exists and to calculate its value if it does:

1. Check to see whether $f$ is a closed form function. These are functions that can be specified by a single formula involving constants, powers of $x,$ radicals, exponentials and logarithms, combined using arithmetic operations and composition of functions. If $f$ is a closed-form function, then:
   1. If $a$ is in the domain of $f,$ then $\displaystyle \lim_{x \to a}f(x)$ exists and equals $f(a).$
   2. If $a$ is not in the domain of $f,$ but $f$ can be reduced by simplification to a closed-form function $g$ that does have $a$ in its domain, then $\displaystyle \lim_{x \to a}f(x)$ exists and equals $g(a).$.
   3. If $a$ is not in the domain of $f,$ and you cannot simplify the function as in (b), then simplify as much as possible, and estimate the limit by further analysis (for instance, if substitution leads to a determinate form like $k/0$).
2. If $f$ is not closed-form, and $a$ is a point at which the definition of $f$ changes, compute the left limit and right limit seperately, and check whether they agree.

Computing limits at infinity algebraically

To decide algebraically whether $\displaystyle \lim_{x \to \pm\infty}f(x)$ exists and to calculate its value if it does:

Check to see whether $f(x)$ is a ratio of polynomials (that is, a rational function). If it is, then you can ignore all but the highest powers of $x$ in the numerator and denominator. This simpler function will have the same limit as $f.$