September « 2008 « The Science That Draws Necessary Conclusions

Archive for September 2008

Intro to Limits

A limit is what f(x) approaches as x approaches a value from both directions. The limit of a function f(x) as x approaches 0 is written as $\displaystyle\lim_{x \to 0}f(x)$ .

f(x) is pictured above. It is evident that $\displaystyle\lim_{x \to 3}f(x)=5$ . As x gets closer to the value 3, f(x) gets closer to 5.

f(x) never needs to equal the number x approaches. $\displaystyle\lim_{x \to 4}g(x)=3$ even though $g(4)=5$ .

If there is a vertical asymptote asymptote at x=0, $\displaystyle\lim_{x \to 0}f(x)=DNE$ . DNE stands for “does not exist”. The line approaching x from the negative side goes to $-\infty$ . From the positive side, the line appproaches $\infty$ . Because they are not approaching the same point, the limit is said not to exist.

Limits can pertain to only the line coming from only one side. It is written as $\displaystyle\lim_{x \to 0^+}f(x)$ if it only pertains to parts of the line where $x>0$ or as $\displaystyle\lim_{x \to 0^-}f(x)$ if it only pertains to parts of the line where $x<0$ . If $f(x)=\frac{1}{x}$ , then $\displaystyle\lim_{x \to 0^+}f(x) = +\infty$ and $\displaystyle\lim_{x \to 0^-}f(x) = -\infty$ .