Changes

=Limits=

=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

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

<math>f(x) = \frac{2x^2-15+7}{2x-1} \ </math>

<math>f(x) = \frac{2x^2-15+7}{2x-1} \ </math>

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

If we factor the [[quadratic forms|quadratic]] in the [[numerator]], we see that <math>2x^2-15x+7=(2x-1)(x-7) \ </math> and hence for almost all values of ''x'', the function is simply <math>f(x)=x-7 \ </math>.   

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

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

<math>\lim_{x\to 1/2} f(x) = \frac{-13}{2} \ </math>.

<math>\lim_{x\to 1/2} f(x) = \frac{-13}{2} \ </math>.