Difference between revisions of "Calc1.2"
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=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>. | ||
==Formal Definition== | ==Formal Definition== | ||
| + | [[Category:Mathematics]] | ||
Latest revision as of 04:53, April 13, 2013
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
If we factor the quadratic in the numerator, we see that and hence for almost all values of x, the function is simply .
But at the crucial point , the denominator and so requires division by zero and hence is undefined. However, since the function satisfies for , it isn't hard to guess by the above informal description of a limit that the limit of f at x=1/2 is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2-7=-13/2 \ } . We write:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to 1/2} f(x) = \frac{-13}{2} \ } .