AP Calculus BC

## What a limit says The limit of $f(x)$ as $x$ approaches $c$ is the number $R$ that $f(x)$ can be made arbitrarily close to by taking $x$ sufficiently close to $c$, without letting $x$ equal $c$. It is written $$ \lim_{x \to c} f(x) = R. $$ The value of $f$ at $c$ plays no part. The function may be undefined at $c$, or defined with a different value, and the limit is still decided only by what happens nearby. That is why limits are the tool for describing the behaviour of $\frac{x^2 - 9}{x - 3}$ at $x = 3$, where the expression itself is not defined. A limit is a real number or it does not exist. Writing $\lim_{x \to c} f(x) = \infty$ describes the way in which a limit fails to exist, and it is not a value. ## One-sided limits $\lim_{x \to c^-} f(x)$ is the limit as $x$ approaches $c$ from the left, using only values below $c$. $\lim_{x \to c^+} f(x)$ uses only values above $c$. The two-sided limit exists exactly when both one-sided limits exist and are equal: $$ \lim_{x \to c} f(x) = R \iff \lim_{x \to c^-} f(x) = R \text{ and } \lim_{x \to c^+} f(x) = R. $$ For $f(x) = 2x - 1$ when $x < 3$ and $f(x) = x^2 - 6$ when $x \ge 3$, the left limit at $3$ is $2(3) - 1 = 5$ and the right limit is $9 - 6 = 3$. They differ, so $\lim_{x \to 3} f(x)$ does not exist, even though $f(3) = 3$ is defined. ## Ways a limit can fail to exist There are three common reasons. - The one-sided limits exist but are different, as in the jump above. A graph shows this as a break with the two sides…