A limit is a concept in mathematics describing what value a [[function]] $f(x)$ approaches as $x$ "goes to" (or approaches) some value. For example, the limit of the function $1/x$ as $x$ goes to infinity is $0$. $\lim_{x \to \infty} \frac1x = 0$ Limits of functions can be calculated for any value including infinity. ## one-sided limit Some functions will have different limits as $x$ approaches some value depending on whether, graphically, you are coming from the left or right. These are called "left limits" and "right limits", respectively. In this case, the limit (in the general sense) does not exist. The notation is $\lim_{x\to m}^+ = a, \lim_{x\to m}^- = b$ where the plus and minus operators indicate the right and left limits, respectively. The limit as a [[piecewise function]] approaches the value at which the function is discontinuous will have a "left limit" and "right limit" which may not agree. For a function that goes to infinity, for example $f(x) = 1/x$, the limit as $x$ goes to infinity is $0$ but the limit as $x$ approaches $0$ does not exist (in this case, the left limit is positive infinity and the right limit is negative infinity). ## limit rules The limit of $n$ as $n \to \infty$ if $n$ is in the denominator is 0. $\lim_{n \to \infty} \frac1n = 0$ The limit of a fraction raised to the power of $n$ is 0. $\lim_{n \to \infty} (\frac12)^n = 0$ If the degree of the degree of the polynomial in the denominator is greater than the degree of the polynomial in the numerator, the limit of $n$ as $n \to \infty$ is $0$. $\lim_{n \to \infty} \frac{2n^2 + 3}{3n^3 - 7} = 0$ If the degree of the degree of the polynomial in the denominator is the same as the degree of the polynomial in the numerator, the limit of $n$ as $n \to \infty$ is the ratio of the lead coefficients. $\lim_{n \to \infty} \frac{2n^2 + 3}{3n^2 - 7} = \frac23$ The limit of a series $a_n$ where $a_n = \frac{n-1}{n}$ converges to $1$ as $n$ goes to infinity.