(a) Determine whether $$ f(x)=\frac{|x-4|-2}{x} $$ is continuous at $$ x=4 $$. [6 marks] (b) Given $$ f(x)=\frac{2 x}{\sqrt{x^{2}+2}} $$. By finding the limit at appropriate values, obtain the horizontal and vertical asymptotes, if they exist. Hence, sketch the graphs of $$ f(x)=\frac{2 x}{\sqrt{x^{2}+2}} $$.

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Parsons Matthews · Accepted Answer

(a) $$ f(x) $$ is continuous at $$ x = 4 $$. (b) The horizontal asymptote is $$ y = 2 $$. The graph starts below $$ y = 2 $$ for negative $$ x $$, crosses $$ y = 2 $$, and approaches it as $$ x $$ increases.

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