# 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} \$.