The notion of limit

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Let \(a\) and \(b\) be real numbers and let \(f\) be a real function that is defined on an open interval containing \(a\).

We say that \(f\) has limit \(b\) at \(a\) if \(f(x)\) comes closer to \(b\) as \(x\) comes closer to \(a\).

In this case, we write \(\textstyle\lim_{x\to a} f(x) = b\) or \(\displaystyle\lim_{x\to a} f(x) = b\).


Let’s practice!

The rational function \(f(x) = \frac{2\cdot x-16}{x-8}\) is defined everywhere except at \(8\).

What is the limit of \(f\) at \(8\)?

For every value of \(x\), \(f(x)=2\) is close to (but distinct from) \(8\).

Answer:

\(\lim_{x\to 8}f(x)= 2\)

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