Section 1.1 Introduction
Use the following definitions, theorems, and properties to solve the problems contained in this Chapter.
- Limit
We write \(\displaystyle \lim _{x\to a}f(x)=L\) and say “the limit of \(f(x)\text{,}\) as \(x\) approaches \(a\text{,}\) equals \(L\)” if it is possible to make the values of \(f(x)\) arbitrarily close to \(L\) by taking \(x\) to be sufficiently close to \(a\text{.}\)
- Limit - \(\mathbf{\varepsilon,\ \delta}\) Definition
Let \(f\) be a function defined on some open interval that contains \(a\text{,}\) except possibly at \(a\) itself. Then we say that the limit of \(f(x)\) as \(x\) approaches \(a\) is \(L\text{,}\) and we write \(\displaystyle \lim _{x\to a}f(x)=L\) if for every number \(\varepsilon >0\) there is a \(\delta >0\) such that \(|f(x)-L|\lt \varepsilon\) whenever \(0\lt |x-a|\lt \delta\text{.}\)
- Limit and Right-hand and Left-hand Limits
\(\displaystyle \displaystyle \lim _{x\to a}f(x)=L\Leftrightarrow (\lim _{x\to a^-}f(x)=L\mbox{ and } \lim _{x\to a^+}f(x)=L)\)
- Infinite Limit
Let \(f\) be a function defined on a neighbourhood of \(a\text{,}\) except possibly at \(a\) itself. Then \(\displaystyle \lim _{x\to a}f(x)=\infty\) means that the values of \(f(x)\) can be made arbitrarily large by taking \(x\) sufficiently close to \(a\text{,}\) but not equal to \(a\text{.}\)
- Vertical Asymptote
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The line \(x=a\) is called a vertical asymptote of the curve \(y=f(x)\) if at least one of the following statements is true:
\begin{equation*} \begin{array}{lll} \displaystyle \lim _{x\to a}f(x)=\infty \amp \displaystyle \lim _{x\to a^-}f(x)=\displaystyle \infty \amp \displaystyle \lim _{x\to a^+}f(x)=\infty \\ \displaystyle \lim _{x\to a}f(x)=-\infty \amp \displaystyle \lim _{x\to a^-}f(x)=-\infty \amp \displaystyle \lim _{x\to a^+}f(x)=-\infty \end{array} \end{equation*} - Limit At Infinity
Let \(f\) be a function defined on \((a,\infty )\text{.}\) Then \(\displaystyle \lim _{x\to \infty }f(x)=L\) means that the values of \(f(x)\) can be made arbitrarily close to \(L\) by taking \(x\) sufficiently large.
- Horizontal Asymptote
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The line \(y=a\) is called a horizontal asymptote of the curve \(y=f(x)\) if if at least one of the following statements is true:
\begin{equation*} \lim _{x\to \infty }f(x)=a \mbox{ or } \lim _{x\to -\infty }f(x)=a\text{.} \end{equation*} - Limit Laws
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Let \(c\) be a constant and let the limits \(\displaystyle \lim _{x\to a}f(x)\) and \(\displaystyle \lim _{x\to a}g(x)\) exist. Then
\(\displaystyle \displaystyle \lim _{x\to a}(f(x)\pm g(x))=\lim _{x\to a}f(x)\pm\lim _{x\to a}g(x)\)
\(\displaystyle \displaystyle \lim _{x\to a}(c\cdot f(x))=c\cdot \lim _{x\to a}f(x)\)
\(\displaystyle \displaystyle \lim _{x\to a}(f(x)\cdot g(x))=\lim _{x\to a}f(x)\cdot \lim _{x\to a}g(x)\)
\(\displaystyle \lim _{x\to a}\frac{f(x)}{g(x)}=\frac{\lim _{x\to a}f(x)}{\lim _{x\to a}g(x)}\) if \(\lim _{x\to a}g(x)\not= 0\text{.}\)
- Squeeze Law
If \(f(x)\leq g(x)\leq h(x)\) when \(x\) is near \(a\) (except possibly at \(a\)) and \(\displaystyle \lim _{x\to a}f(x)=\lim _{x\to a}h(x)=L\) then \(\displaystyle \lim _{x\to a}g(x)=L\text{.}\)
- Trigonometric Limits
\(\displaystyle \lim_{\theta \to 0}\frac{\sin{\theta}}{\theta}=1\) and \(\displaystyle \lim_{\theta \to 0}\frac{\cos{\theta}-1}{\theta}=0\text{.}\)
- The Number \(\mathbf{e}\)
\(\displaystyle \lim_{x \to 0}(1+x)^{\frac{1}{x}}=e\) and \(\displaystyle \lim_{x \to \infty }\left( 1+\frac{1}{x}\right) ^x=e\text{.}\)
- L'Hospital's Rule
Suppose that \(f\) and \(g\) are differentiable and \(g'(x)\not= 0\) near \(a\) (except possibly at \(a\text{.}\)) Suppose that \(\ds \lim _{x\to a}f(x)=0\) and \(\ds \lim _{x\to a}g(x)=0\) or that \(\ds \lim _{x\to a}f(x)=\pm \infty\) and \(\ds \lim _{x\to a}g(x)=\pm \infty\text{.}\) Then \(\ds \lim _{x\to a}\frac{f(x)}{g(x)}=\lim _{x\to a}\frac{f'(x)}{g'(x)}\) if the limit on the right side exists (or is \(\infty\) or \(-\infty\)).
- Continuity
We say that a function \(f\) is continuous at a number \(a\) if \(\displaystyle \lim _{x\to a}f(x)=f(a)\text{.}\)
- Continuity and Limit
If \(f\) is continuous at \(b\) and \(\displaystyle \lim _{x\to a}g(x)=b\) then \(\displaystyle \lim _{x\to a}f(g(x))=f(\lim _{x\to a}g(x))=f(b)\text{.}\)
- Intermediate Value Theorem
Let \(f\) be continuous on the closed interval \([a,b]\) and let \(f(a)\not= f(b)\text{.}\) For any number \(M\) between \(f(a)\) and \(f(b)\) there exists a number \(c\) in \((a,b)\) such that \(f(c)=M\text{.}\)