| utf03-004 Given the graph of $ f( x)$, provide an expression for the following functions $ h(x)$, $ g(x)$, and $ v(x)$ in terms of $ f( x)$. {(A function $ u(x)$, say, written in terms of $ f( x)$ would be $ u(x) = a f( {bx+c}) + d,$ where $a, b, c, $ and $d$ are constants.)} |
| utf03-006 Which of the graphs below could correspond to the following functions? {Match them.} \begin{alignat*}{4} &a) \text{odd power function}\hphantom{\text{log x}} & &b) \text{ even power function} & &c) \text{a quadratic}\\ &d) \log x & &e) \text{a 4th degree polynomial} & &f) b^x \\ &g) \sin x & &h) (\frac{1}{ 2})^x & &i) \text{absolute value} \end{alignat*} |
| utf05-001 Sketch the functions $g(x), h(x), $ and $ f( x)$ such that the following conditions hold. \begin{align*} a) &\lim_{x \to 5}{g(x)} = 10 \text{however} g(5) \text{is not defined.}\hskip2truein\\ b) &\lim_{x \to 2}{h(x)} = 3 \text{however} h(2)\neq 3.\\ c) &\lim_{x \to 0}{ f( x)} \text{does not exist, however} f( 0) = 2. \end{align*} |
| utf05-005 For each below do the following: Compute the limits, write in words what techniques you used to solve them, then write another limit of the same type. \begin{alignat*}{2} a) &\lim_{x \to 1}\frac{x^3-x^2+x-1 }{ x^2-1} & &\lim_{x\to 0}\frac{(1+x)^2-1 }{ (1+x)^3 - 1} \\ b) &\lim_{x \to 4}\frac{\root \uproot 3 3 \of {x} - \root \uproot 3 3 \of {4} }{ x-4} & &\lim_{x \to 0}\frac{(x-2) + 2 }{ {\root \uproot 3 3 \of {x-2} + \root \uproot 3 3 \of {2}}} \\ c) &\lim_{x \to 3}\frac{2x^3-18x^2+54x-54 }{ x-3} & &\lim_{x \to -1}\frac{x^4+ 4x^3+6x^2+4x+1 }{ (x^2-1)^4} \\ d) &\lim_{x \to 3}\frac{\sqrt x - \sqrt 3 }{ x-3} & &\lim_{x \to 0}\frac{x^2 }{ 1 - \sqrt {1-x^2}} \\ e) &\lim_{x \to a}\frac{|x-a| }{ x-a} & &\lim_{x \to -3^{-}}\frac{|x+3| }{ x+3} \\ f) &\lim_{x \to 1}\frac{\root 3 \of x - 1 }{ x-1} & &\lim_{x \to 1}\frac{\root 4 \of x -1 }{ x-1} \\ g) &\lim_{x \to 2} f( x), \lim_{x \to 2^{-}} f( x), \lim_{x \to 2^{+}} f( x) & & \text{where} f( x) = \begin{cases} x^2,& x \geq 2 \\ \frac{x^2-4 }{ x-2},& x<2 \end{cases} \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 $\infty \cdot 0$? |