| utf06-002 Compute the limits. \begin{alignat*}{3} &a) \lim_{x \to 0} {\frac{\sin^2{2x}}{ x^2}} & &b) \lim_{x \to 0}{\frac{1-\cos{3x} }{ x^2}} & &c) \lim_{x \to 0}{\frac{x}{ 1-\sec x}} \\ &d) \lim_{x \to 0}{\frac{1+\cot^2 {3x} }{ 4x^2}} & &e) \lim_{x\to \frac{\pi }{ 6}} {[ 1-\sec {(x+\frac{\pi }{ 3})}] (x - \frac{\pi }{ 6})^2} & &f) \lim_{x\to \frac{\pi }{ 2}}{\frac{\cos x }{ 2x - \pi}} \end{alignat*} |
| utf06-003 Find an appropriate $ f( x)$ for each of the following. \begin{alignat*}{6} a) &\lim_{x \to 2}{ f( x)} &&= 0, & f( 2) &=\frac{0}{ 0} & b) &\lim_{x \to -1}{ f( x)}&&= 2, & f( {-1}) &=\frac{0}{ 0} \\ c) &\lim_{x \to \pi}{ f( x)} &&= \infty, & f( {\pi}) &= \infty \cdot 0 & d) &\lim_{x \to 1}{ f( x)} &&= 1, & f( {1}) &=\frac{0}{ 0} \\ e) &\lim_{x \to \pi}{ f( x)} &&= 0, & f( {\pi}) &= \infty \cdot 0 & f) &\lim_{x \to d}{ f( x)} &&= 3, & f( {d}) &=\infty \cdot 0 \end{alignat*} Do you notice anything strange about the limit of $ f( x)$ as $x \to c$ when $ f( c)$ has the form $\infty \cdot 0$? |
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| utf06-007 Show that if $0\leq f( x) \leq 1$ for each $x\in [0,1] $ and $ f( x)$ is a continuous function, then there is some number $a$, $0\leq a \leq 1$, such that $ f( a) = a$. |
| utf06-008 More limits! \begin{alignat*}{3} &a) \lim_{x \to 0}{\frac{\tan {3x} }{ x}} & &b) \lim_{x \to 0} {\frac{\sec {2x} \tan {2x} }{ x}} & &c) \lim_{t \to 0^{+}} \sin (\sqrt x \cdot x)\\ &d) \lim_{x \to 0}{\frac{1-\cos {2x} }{ x}} & &e) \lim_{x \to 0}{\frac{1-\cos {2x} }{ x^2}} \end{alignat*} |