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'''3.''' Use [[mathematical induction]] to show that <math>\lim_{x \rightarrow x_0} a_n x^n + ... + a_1 x + a_0 = a_n x_0^n + ... + a_0 \ </math>.   

'''3.''' Use [[mathematical induction]] to show that <math>\lim_{x \rightarrow x_0} a_n x^n + ... + a_1 x + a_0 = a_n x_0^n + ... + a_0 \ </math>.   

You may assume that for any functions <math>A, B, \lim_{x \rightarrow x_0} (A(x)+B(x)) = \lim_{x \rightarrow x_0} A(x)+ \lim_{x \rightarrow x_0} B(x) \ </math> and <math>\lim_{x \rightarrow x_0} (A(x)B(x)) = \lim_{x \rightarrow x_0} A(x) \times \lim_{x \rightarrow x_0} B(x) \ </math>.

You may assume that for any functions with valid limits <math>A, B, \ </math>

 

<math>\lim_{x \rightarrow x_0} (A(x)+B(x)) = \lim_{x \rightarrow x_0} A(x)+ \lim_{x \rightarrow x_0} B(x) \ </math>  

 

and <math>\lim_{x \rightarrow x_0} (A(x)B(x)) = \lim_{x \rightarrow x_0} A(x) \times \lim_{x \rightarrow x_0} B(x) \ </math>.

'''4.''' Using the definition of "limit" given above, find <math>\lim_{x \rightarrow 0} \frac{\sin(x)}{x} \ </math>.

'''4.''' Using the definition of "limit" given above, find <math>\lim_{x \rightarrow 0} \frac{\sin(x)}{x} \ </math>.