# Limit **A limit of $f(x)$ at $x=a$ is said to exist if both a left limit and a right limit exists at that point, and their values are equal.** In this case $f(x)$ is said to be continuous at $a$. Otherwise a [[Function (in Algebra)#Function Discontinuities|discontinuity]] is said to exist at $a$. ## Limit Notation The following denotes a limit of $f(x)$ at $x=a$. $\lim_{x \to a} {f(x)} = L$ ## Limit Rules $\lim_{x \to a} c = c$ ![[The Limit Constant Rule#^2uw6gl]] (where $c$ is any constant) ![[The Limit Sum Rule#^wjie2a]] The product rule: $\lim_{x \to a} {[f(x) \cdot g(x)]} = \lim_{x \to a} {f(x)} \cdot \lim_{x \to a} {g(x)}$ ![[The Limit Power Rule#^f2joqi]] The chain rule: $\lim_{x \to a} {[f(g(x))]} = \lim_{x \to a} {f (\lim_{x \to a} {g(x)})}$ For any polynomial $p(x)$: ![[The Limit Polynomial Rule#^2f957s]] L'Hôpital's Rule: ![[L'Hopital's Rule#^853loo]] ## Tactics for Evaluating Limits To evaluate $\lim f(x)$ first evaluate $f(x)$. Three cases exist: 1. If you get a constant the limit is the constant, unless you're dealing with a piecewise function in which case you need to make sure both the left and right limits exist and are equal 2. $f(x) = \frac c 0$ in which case the limit does not actually exist, although some would say that the limit as $x$ goes to $a$ is $\infty$. 3. $f(x) = \frac 0 0$ or $\frac \infty \infty$ a.k.a. the indeterminate form, which happens when both the numerator and the denominator of a fraction are approaching $0$ as $x$ is approaching $a$. The limit then depends on how fast the numerator and denominator are approaching $0$ relative to each other. If this happens the function has a discontinuity at $x=a$ and the limit must be determined by applying algebraic manipulation of $f$, keeping in mind that the domain of the newly manipulated function does NOT have to match the domain of $f$ e.g. it's perfectly fine to cancel $x$ in $\frac {x^2}{x}$. - If dealing with fractions where both the numerator and denominator are polynomials, it's often helpful to divide both with the highest power of $x$. For [[Trigonometric Functions|Trig Functions]] see [[The Squeeze Theorem#Using The Squeeze Theorem To Evaluate Limits of Trigonometric Functions]]. ### Evaluating Limits When $x \to \infty$ When trying to evaluate a limit of the form $\lim_{x \to \infty} \frac {p(x)}{q(x)}$ where $p$ and $q$ are non-zero polynomials, the limit is: - $\infty$ or $-\infty$, if the degree of $p$ is bigger than the degree of $q$ - 0, if the degree of $p$ is smaller than the degree of $q$ - The degrees are equal, the limit is the ratio of the leading coefficients Another thing to consider is if one of $p$ or $q$ are exponential ($f(x) = a^x$). If so then $f(x)$ will approach infinity faster. ## Common Limits ![[The Natural Constant#^m74vgi]] ![[The Natural Constant#^rahqyz]] $\begin{align} &\lim_{n \to \infty} \frac {P(n)} {e^n} = 0 && \text {where $P$ is any polynomial} \\ &\lim_{n \to \infty} \sqrt[n]n = 1 \\ &\lim_{n \to \infty} \sqrt[n] {n!} = +\infty \\ &\lim_{n \to \infty} \arctan(n) = \frac \pi 2 \\ &\lim_{n \to \infty} a^n = 0 && \text {when } {-1} < a < 1 \\ &\lim_{n \to \infty} \frac {\ln(n)} {n^a} = 0 && \text {when } a > 0 \\ &\lim_{n \to \infty} \sqrt[n] a = 1 && \text {when } a > 0 \\ &\lim_{n \to \infty} \frac {A^n} {n!} = 0 && \text{for any $A$ } \end{align} $ ## Rigorous Definition $\lim_{x \to a} f(x) = L$ is said to exist if: if given any $\epsilon > 0$, one could come up with a positive value $\delta$ such that for all $x$ values within the range $a - \delta < x < a + \delta$ the following holds: $|f(x) - L| < \epsilon$. In plain English: a limit exists if no matter how close I want to get to $L$, you can come up with a whole range of $x$ inputs that will get me still closer.