4.1 Intuitive definition of limit

All three functions above are undefined at \(x = 2\) (we cannot evaluate \(f(2), g(2), h(2)\)). But we can evaluate them neat the point \(x = 2\) and these evaluations will be very different.

By visual exploration of graph (a) we see that \(\lim_{x \to 2} f(x) = 4\).

Definition: Let \(f(x)\) be a function defined on an open interval around point \(a\) and \(L\) a real number. If \(f(x) \to L\) (the values \(f(x)\) approach \(L\)) as \(x \to a\) (\(x\) approaches \(a\)) then we say the limit as \(x\) approaches \(a\) is \(L\). \[\lim_{x \to a} f(x) = L\]

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Limit of a function makes sense even for points where the function is defined.

Careful: For most points \(a\) within the domain of the function we have \(\lim_{x \to a} f(x) = f(a)\). However, in some cases we may have \(\lim_{x \to a} f(x) \neq f(a)\) as for the function \(g\) in the image below.

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Two important limit rules:

• \(\lim_{x \to a} x = a\)
• \(\lim_{x \to a} c = c\)

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Careful: The limit of a funciton \(f(x)\) at a point \(a\) exists only if the values of the function approach a single real number as \(x \to a\).

Sometimes, the limit from the right \[\lim_{x \to a^+} f(x)\] as \(x\) approaches \(a\) with \(x > a\) is not equal to the limit from the left \[\lim_{x \to a^-} f(x)\] as \(x\) approaches \(a\) with \(x < a\).

In such a case the limit does not exist and we can only speak about one-sided limits.

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In some cases the function \(f(x)\) does not approach a real number \(L\) as it nears the point \(a\) but may increase or decrease without a bound.

We speak about infinite limits (left, right or two-sided) \[\lim_{x \to a} f(x) = \pm \infty\]

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The line \(x = a\) for a function \(f(x)\) with an ifinite limite at point \(a\) is called a vertical asymptote.

4.2 The limit laws

Two simple limit rules for a real number \(a\) and a constant \(c\) \[\begin{eqnarray} \lim_{x \to a} x & = & a \\ \lim_{x \to a} c & = & c \end{eqnarray}\]

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Let \(f(x)\) and \(g(x)\) be defined on a open interval around \(a\) for all \(x \neq a\), the real numbers \(L, M\) be the limits \(\lim_{x \to a} f(x) = L\) and \(\lim_{x \to a} g(x) = M\), and \(c\) be constant. Then these statments hold: \[\begin{eqnarray} \lim_{x \to a} \big( f(x) \pm g(x) \big) & = & \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) = L \pm M\\ \lim_{x \to a} c f(x) & = & c \lim_{x \to a} f(x) = c L\\ \lim_{x \to a} \big( f(x) \times g(x) \big) & = & \lim_{x \to a} f(x) \times \lim_{x \to a} g(x) = L \times M\\ \lim_{x \to a} \frac{f(x)}{g(x)} & = & \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{L}{M} \quad \text{for } M \neq 0\\ \lim_{x \to a} \big(f(x)\big)^n & = & \big( \lim_{x \to a} f(x) \big)^n = L^n \quad \text{for all } n \text{ positive integers}\\ \lim_{x \to a} \sqrt[n]{f(x)} & = & \sqrt[n]{\lim_{x \to a} f(x)} = \sqrt[n]{L} \quad \text{for all } L \text{ if } n \text{ is odd or for } L \leq 0 \text{ if } n \text{ is even} \\ \lim_{x \to a} f\big( g(x) \big) & = & f \big( \lim_{x \to a} g(x) \big) = f(M) \end{eqnarray}\]

Hint: If for the fraction fraction \(\frac{f(x)}{g(x)}\) the limits of the terms yield the indeterminate form \[\frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} = \frac{0}{0}\] we should try to factor the polynomials in \(f(x)\) and \(g(x)\) to first simplify and then calculate the limit.

4.3 Continuity

The graphs of continous functions can be traced without lifting a pencil. When there is a break, the functions are discontinous at that point.

Definition: A function \(f(x)\) is continuous at a point \(a\) if

1. \(f(a)\) is defined
2. \(\lim_{x \to a} f(x)\) exists
3. \(\lim_{x \to a} f(x) = f(a)\)

A function is discontinous at point \(a\) otherwise.

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A function \(f(x)\) is continous from the right at point \(a\) if \(\lim_{x \to a^+} f(x) = f(a)\) and continous from the left if \(\lim_{x \to a^-} f(x) = f(a)\).

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A function \(f(x)\) is continuous on an open interval \((a, b)\) if it is continuous at every point \(x \in (a, b)\).

A function \(f(x)\) is continuous on a closed interval \([a, b]\) if it is continuous at every point \(x \in (a, b)\) and continous from the right at \(a\) and from the left at \(b\).

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Intermediate value theorem: If \(f(x)\) is continuous over closed interval \([a, b]\) the for any real number \(z\) between \(f(a)\) and \(f(b)\) there exists a number \(c \in [a, b]\) such that \(f(c) = z\).

4.4 Formal definition of limit

The epsilon-delta definition of a limit: Let \(f(x)\) be a function defined at all points \(x \neq a\) around \(a\) and \(L\) a real number. Then \[\lim_{x \to a} f(x) = L\] if for every \(\epsilon > 0\) there exists \(\delta > 0\) such that if \(0 < |x - a| < \delta\), then \(|f(x) - L| < \epsilon\).

Limit from the right: Let \(f(x)\) be a function defined at all points \(x \neq a\) around \(a\) and \(L\) a real number. Then \[\lim_{x \to a^+} f(x) = L\] if for every \(\epsilon > 0\) there exists \(\delta > 0\) such that if \(0 < x - a < \delta\), then \(|f(x) - L| < \epsilon\).

Limit from the left: Let \(f(x)\) be a function defined at all points \(x \neq a\) around \(a\) and \(L\) a real number. Then \[\lim_{x \to a^-} f(x) = L\] if for every \(\epsilon > 0\) there exists \(\delta > 0\) such that if \(-\delta < x - a < 0\), then \(|f(x) - L| < \epsilon\).

Infinite limite: Let \(f(x)\) be a function defined at all points \(x \neq a\) around \(a\). Then \[\lim_{x \to a} f(x) = \infty\] if for every \(M > 0\) there exists \(\delta > 0\) such that if \(-\delta < x - a < 0\), then \(f(x) > M\).