2.3: The Limit Laws
- Page ID
- 143299
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Evaluate \(\displaystyle \lim_{x \to 2} 2x^2\), if possible.
- Evaluate \(\displaystyle \lim_{x \to -3} 7\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 5} (3x - 12)\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 2} \dfrac{x+7}{x-3}\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 1} \dfrac{\ln x}{\cos(\pi x)}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to -1} \sqrt{x^2 - 7x - 2}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to 0} \dfrac{x}{x}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to -3} \dfrac{x^2 - 9}{x + 3}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to 4} \dfrac{x^2 - 6x + 8}{x^2 + x - 20}\), if possible.
- Evaluate \( \displaystyle \lim_{h \to 0} \dfrac{(x + h)^2 - x^2}{h} \), if possible.
- Evaluate \( \displaystyle \lim_{t \to 4} \dfrac{\sqrt{t} - 2}{t - 4} \), if possible.
- Evaluate \( \displaystyle \lim_{\theta \to \pi} \dfrac{\sin \theta}{\tan \theta}\), if possible.
- Evaluate \(\displaystyle \lim_{h \to 0} \dfrac{\dfrac{1}{x+h} - \dfrac{1}{x}}{h}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to 5} \dfrac{x - 5}{3 - \sqrt{x + 4}} \), if possible.
- Evaluate \(\displaystyle \lim_{x \to 0} \sin\left(\dfrac{1}{x}\right)\), if possible.
- Evaluate \( \displaystyle \lim_{x \to -2^-} \dfrac{x^2-1}{x^2-x-6}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to -2^+} \dfrac{x^2-1}{x^2-x-6}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to -2} \dfrac{x^2-1}{x^2-x-6}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to 3^-} \dfrac{x^2-1}{x^2-x-6}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to 3^+} \dfrac{x^2-1}{x^2-x-6}\), if possible.
- Evaluate \( \displaystyle \lim_{x \to 3} \dfrac{x^2-1}{x^2-x-6}\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 1^-} \dfrac{1}{x^2 - 2x + 1}\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 1^+} \dfrac{1}{x^2 - 2x + 1}\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 1} \dfrac{1}{x^2 - 2x + 1}\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 0^+} \sqrt{x}\), if possible.
- Evaluate \(\displaystyle \lim_{x \to 0^+} \ln x\), if possible.
- Consider the function
\[ f(x) = \left\{\begin{align*}
& x^2, & x < 1, \\
& x, & x \ge 1.
\end{align*}\right.\]
Find \(\displaystyle\lim_{x \to 1^-} f(x)\), \(\displaystyle\lim_{x \to 1^+} f(x)\), and \(\displaystyle\lim_{x \to 1} f(x)\), if possible.
- Consider the function
\[ g(x) = \left\{\begin{align*}
& \cos x, & x \le 0, \\
& \sin x, & x > 0.
\end{align*}\right.\]
Find \(\displaystyle\lim_{x \to 0^-} g(x)\), \(\displaystyle\lim_{x \to 0^+} g(x)\), and \(\displaystyle\lim_{x \to 0} g(x)\), if possible.
- Consider the function
\[ h(x) = \left\{\begin{align*}
& \dfrac{1}{x+1}, & x < -1, \\
& \dfrac{1}{x}, & -1 \le x < 0.
\end{align*}\right.\]
Find \(\displaystyle\lim_{x \to -1^-} h(x)\), \(\displaystyle\lim_{x \to -1^+} h(x)\), and \(\displaystyle\lim_{x \to -1} h(x)\), if possible.
- Consider the function
\[ f(x) = \left\{\begin{align*}
& |1 - x^2|, & x < 0, \\
& \sqrt{2}, & x = 0, \\
& e^x, & x > 0.
\end{align*}\right.\]
Find \(\displaystyle\lim_{x \to 0^-} f(x)\), \(\displaystyle\lim_{x \to 0^+} f(x)\), and \(\displaystyle\lim_{x \to 0} f(x)\), if possible.