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*}
– James Epperson 1998-08-09