2.3 Properties of Limits

1.: Uniqueness of a limit Suppose \(f(x) \longrightarrow L_1\) as \(x \longrightarrow c\) and \(f(x) \longrightarrow L_2\) as \(x\longrightarrow c\). Then \(L_1 = L_2\).
2.: Limit of a constant If \(K\) is a constant and \(f(x) = K\hspace {0.3cm} \forall \) values of \(x\). Then for any number \(c\), \(\lim \limits _{x \rightarrow c} f(x) = K\)
3.: If \(c\) is a real number and \(f(x) = x\) then \(\displaystyle {\lim \limits _{x\rightarrow c} f(x) = c}\)
4.: Limits of equal functions Suppose that there is a number \(h>0\) such that \(f(x) = g(x)\hspace {0.2cm} \forall x\) for which \(\left |x - c\right | < h\). Suppose also that \(\lim \limits _{x\rightarrow c} f(x) = L\), then \(\lim \limits _{x\rightarrow c} g(x) = L\).
5.: Limit of a sum If \(f\) and \(g\) are two functions with \(\lim \limits _{x\rightarrow c} f(x) = L_1\) and \(\lim \limits _{x\rightarrow c} g(x) = L_2\) then
\(\lim \limits _{x\rightarrow c} \Big [ f(x) + g(x)\Big ] = L_1 +L_2\)
6.: Limit of a product If \(f\) and \(g\) are two functions with \(\lim \limits _{x\rightarrow c} f(x) = L_1\) and \(\lim \limits _{x\rightarrow c} g(x) = L_2\) then
\(\lim \limits _{x\rightarrow c} \big [f(x)\cdot g(x)\big ] = L_1\cdot L_2\)
7.: Limit of a quotient If \(f\) and \(g\) are two functions with \(\lim \limits _{x\rightarrow c} f(x) = L_1\) and \(\lim \limits _{x\rightarrow c} g(x) = L_2\), then
\(\lim \limits _{x\rightarrow c}\dfrac {f(x)}{g(x)} = \dfrac {L_1}{L_2}\), provided \(L_2\neq 0\)
8.: Limit of a composite function Suppose that \(f\) and \(g\) are functions \(b\) and \(c\) are numbers \(f(b)\) is defined and
\(\lim \limits _{x\rightarrow c} f(x) = f(b),\hspace {0.3cm}\lim \limits _{x\rightarrow c} g(x) = b\). Then \(\lim \limits _{x\rightarrow c} f\big [g(x)\big ] = f(b)\)
9.: If \(n\) is a positive integer and \(c > 0\) then \(\displaystyle {\lim \limits _{x\rightarrow c} \sqrt [n]{x} = \sqrt [n]{c}}\)
10.: If \(n\) is positive integer, \(L>0\) and \(\lim \limits _{x\rightarrow c} f(x) = L\), then \(\lim \limits _{x\rightarrow c} \sqrt [n]{f(x)} = \sqrt [n]{L}\)

Example 2.3.1. Let \(f(x) = \sqrt {2x^2 + 1}\) and \(g(x) = 4x - 5\), find \(\lim \limits _{x\rightarrow 2} f(g(x))\).

Solution.

Note that \(\lim \limits _{x\rightarrow 2} \sqrt {2x^2 + 1} = 3\) \begin {align*} \lim \limits _{x\rightarrow 2} f(4x - 5) & = \sqrt {2(4x - 5)^2 + 1} = \sqrt {2(3)^2 + 1}\\ & = \sqrt {19} \end {align*}

\(\lim \limits _{x\rightarrow 2} f(g(x)) = f(3)\)

\(f(3) = \sqrt {2(3)^2 + 1} = \sqrt {19}\)

\(\lim \limits _{x\rightarrow 2} f(g(x)) = f(3) = \sqrt {19}\)

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