Calculus – Limits and continuity

Definition (intuitive): Suppose that \(f(x)\) is defined for \(x\) near \(a\), which means that \(f\) is defined on an open interval that contains \(a\) except possibly \(a\) itself. Then

\[\lim\limits_{x\to a}f(x)=L\]

means that \(f(x)\) comes as close to \(L\) as desired if we choose \(x\) close (enough) to \(a\), but not equal to \(a\).

Notations for one-sided limits: \(\lim\limits_{x\to a^-}f(x)=\lim\limits_{x\uparrow a}f(x)\) (left limit) and \(\lim\limits_{a\to a^+}f(x)=\lim\limits_{x\downarrow a}f(x)\) (right limit), which are defined similarly with \(x < a\) and \(x > a\) respectively.

\[\lim\limits_{x\to a}f(x)=L\quad\Longleftrightarrow\quad\lim\limits_{x\uparrow a}f(x)=L=\lim\limits_{x\downarrow a}f(x).\]

Definition (precise): Suppose that \(f(x)\) is defined on an open interval that contains \(a\) except possibly \(a\) itself. Then

\[\lim\limits_{x\to a}f(x)=L\]

if for every number \(\epsilon > 0\) there exists a number \(\delta > 0\) such that

\[|f(x)-L| < \epsilon\quad\textrm{if}\quad 0 < |x-a| < \delta.\]

Similarly for one-sided limits.

Infinite limits

Sometimes we use the notation \(\lim\limits_{x\to a}f(x)=\infty\) if the values of \(f(x)\) can be made arbitrarily large by taking \(x\) sufficiently close to \(a\), but not equal to \(a\).
Similarly \(\lim\limits_{x\to a}f(x)=-\infty\) means that the values of \(f(x)\) can be made arbitrarily large negative by taking \(x\) sufficiently close to \(a\), but not equal to \(a\).

The squeeze theorem

Theorem: If \(f(x)\leq g(x)\leq h(x)\) when \(x\) is near \(a\), except possibly at \(a\), and

\[\lim\limits_{x\to a}f(x)=L=\lim\limits_{x\to a}h(x),\]

then

\[\lim\limits_{x\to a}g(x)=L.\]

Continuity

Definition: A function \(f\) is continuous at \(x=a\) if \(\lim\limits_{x\to a}f(x)=f(a)\).
Otherwise, \(f\) is said to be discontinuous at \(x=a\). In that case \(f\) has a discontinuity at \(x=a\).

Definition: A function \(f\) is continuous from the left at \(x=a\) if \(\lim\limits_{x\uparrow a}f(x)=f(a)\) and \(f\) is continuous from the right at \(x=a\) if \(\lim\limits_{x\downarrow a}f(x)=f(a)\).

Definition: A function \(f\) is continuous on an interval if it is continuous at every point in the interval. If \(f\) is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the left or continuous from the right.

Theorem: If \(f\) is continuous at \(b\) and \(\lim\limits_{x\to a}g(x)=b\), then \(\lim\limits_{x\to a}f(g(x))=f\left(\lim\limits_{x\to a}g(x)\right)=f(b)\).

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Last modified on October 26, 2021