## University Calculus: Early Transcendentals (3rd Edition)

The limit of a function $f(x)$ as $x\to c$ may fail to exist if: 1) The values of $f(x)$ "jumps" at $x=c$ 2) $f(x)$ approaches $\infty$ as $x\to c$ 3) $f(x)$ oscillates as $x\to c$
The limit of a function $f(x)$ as $x\to c$ may fail to exist if: 1) The values of $f(x)$ "jumps" at $x=c$ Consider this function: $f(x)=0$ for $x\le0$ and $f(x)=1$ for $x\gt0$ As $x$ approaches $0$ from the left, $f(x)$ approaches $0$, but as $x$ approaches $0$ from the right, $f(x)$ approaches $1$. That means $f(x)$ does not approach any single value as $x\to0$, so its limit does not exist. 2) $f(x)$ approaches $\infty$ as $x\to c$ Approaching infinity means the limit does not exist, since it does not reach for any concrete value. Consider the limit: $\lim_{x\to1}\frac{1}{x-1}$ As $x\to1$, $(x-1)$ approaches $0$, so $1/(x-1)$ will get infinitely large, or approach infinity. The limit fails to exist here. 3) $f(x)$ oscillates as $x\to c$ Consider this limit: $\lim_{x\to0}\sin\frac{1}{x}$ A drawing of the graph of the function $y=\sin(1/x)$ would reveal the mad oscillation of $f(x)$ around $x=0$. $f(x)$ does not approach any single value, so the limit fails to exist.