Question

1. Evaluate the following limits. Show all your work to justify each solution. (a) lim_{x?1?} (x?-1)/(x-1) (b) lim_{x?1} (x³-x²+x-1)/(x²-1) (c) lim_{x?0} (?(4-x²)-x)/x² (d) lim_{h?0} (?(a+h)-? a)/h 2. Is it possible that lim_{x?a} f(x) does not exit, and lim_{x?a} g(x) does not exist, but lim_{x?a} [f(x) + g(x)] does exist? Consider the functions f(x) = 1/x and g(x) = -1/x. (a) Sketch the graphs of f(x) and g(x) and find their respective limits as x approaches zero. (b) Sketch the graph of [f(x) + g(x)], and find lim_{z?0} [f(x) + g(x)]. (c) Does your answer in part (b) contradict the Sum Law of Limits? Explain. 3. Sketch the graph of the following piece-wise defined functions f and g and find the following limits. g(x) = { 3 if x < -2; -x - 1 if -2 ? x < 1; x - 5 if 1 ? x < 3 (a) lim_{x?-2?} g(x) (b) lim_{x?2?} g(x) (c) lim_{x?-2} g(x)

          1. Evaluate the following limits. Show all your work to justify each solution.
(a) lim_{x?1?} (x?-1)/(x-1) (b) lim_{x?1} (x³-x²+x-1)/(x²-1) (c) lim_{x?0} (?(4-x²)-x)/x² (d) lim_{h?0} (?(a+h)-? a)/h
2. Is it possible that lim_{x?a} f(x) does not exit, and lim_{x?a} g(x) does not exist, but lim_{x?a} [f(x) + g(x)] does exist?
Consider the functions f(x) = 1/x and g(x) = -1/x.
(a) Sketch the graphs of f(x) and g(x) and find their respective limits as x approaches zero.
(b) Sketch the graph of [f(x) + g(x)], and find lim_{z?0} [f(x) + g(x)].
(c) Does your answer in part (b) contradict the Sum Law of Limits? Explain.
3. Sketch the graph of the following piece-wise defined functions f and g and find the following limits.
g(x) = { 3 if x < -2; -x - 1 if -2 ? x < 1; x - 5 if 1 ? x < 3
(a) lim_{x?-2?} g(x) (b) lim_{x?2?} g(x) (c) lim_{x?-2} g(x)

1. Evaluate the following limits. Show all your work to justify each solution.
(a) limx?1? (x?-1)/(x-1) (b) limx?1 (x³-x²+x-1)/(x²-1) (c) limx?0 (?(4-x²)-x)/x² (d) limh?0 (?(a+h)-? a)/h
2. Is it possible that limx?a f(x) does not exit, and limx?a g(x) does not exist, but limx?a [f(x) + g(x)] does exist?
Consider the functions f(x) = 1/x and g(x) = -1/x.
(a) Sketch the graphs of f(x) and g(x) and find their respective limits as x approaches zero.
(b) Sketch the graph of [f(x) + g(x)], and find limz?0 [f(x) + g(x)].
(c) Does your answer in part (b) contradict the Sum Law of Limits? Explain.
3. Sketch the graph of the following piece-wise defined functions f and g and find the following limits.
g(x) = 3 if x < -2; -x - 1 if -2 ? x < 1; x - 5 if 1 ? x < 3
(a) limx?-2? g(x) (b) limx?2? g(x) (c) limx?-2 g(x)

Added by Lindsey G.

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