Calc1.2

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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} \ .

Formal Definition

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