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Chapter 1: Problem 49
Finding a Limit In Exercises \(47-62,\) find the limit. $$\lim _{x \rightarrow 0} \frac{7 x^{3}-x^{2}}{x}$$
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Continuity of a Composite Function In Exercises \(65-70\) , discuss the continuity of the composite function $$f(x)=\sin x$$ $$g(x)=x^{2}$$
True or False? In Exercises \(105-110\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If $$\lim _{x \rightarrow c} f(x)=L$$ and \(f(c)=L,\) then \(f\) is continuous at \(c\)
Writing Functions Write a function of each pecified type that has a limit of 4 as \(x\) approaches \(8 .\) (a) linear (b) polynomial of degree 2 (c) rational (d) radical (e) cosine (f) sine
Testing for Continuity In Exercises \(75-82,\) describe the interval(s) on which the function is continuous. $$f(x)=\left\\{\begin{array}{ll}{2 x-4,} & {x \neq 3} \\ {1,} & {x=3}\end{array}\right.$$
Signum Function The signum function is defined by $$\operatorname{sgn}(x)=\left\\{\begin{array}{ll}{-1,} & {x<0} \\ {0,} & {x=0} \\ {1,} & {x>0}\end{array}\right.$$ Sketch a graph of \(\operatorname{sgn}(x)\) and find the following (if possible). $$(a) lim _{x \rightarrow 0} \operatorname{sgn}(x) \quad \text { (b) } \lim _{x \rightarrow 0^{+}} \operatorname{sgn}(x) \quad \text { (c) } \lim _{x \rightarrow 0} \operatorname{sgn}(x)$$
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