Calc1.2

Limits

This lecture will introduce a new concept, the limit. A limit, essentially, is what a function "should be" for a certain input. Let's look at an example. Consider the function

$f(x)={\frac {2x^{2}-15+7}{2x-1}}\$

If we factor the quadratic in the numerator, we see that $2x^{2}-15x+7=(2x-1)(x-7)\$ and hence for almost all values of x, the function is simply $f(x)=x-7\$.

But at the crucial point $x=1/2\$, the denominator $2x-1=0\$ and so $f(1/2)\$ requires division by zero and hence is undefined. However, since the function satisfies $f(x)=x-7\$ for $x\neq 1/2\$, it isn't hard to guess by the above informal description of a limit that the limit of f at x=1/2 is $1/2-7=-13/2\$. We write:

$\lim _{x\to 1/2}f(x)={\frac {-13}{2}}\$.