Mathematics

Problem

Consider the function

h(x)=\dfrac{-2}{x-3}

. a. (i) Find the domain and the range of

h

. b. (ii) Calculate

\displaystyle \lim_{x\to3^+} h(x)

and

\displaystyle \lim_{x\to3^-} h(x)

as well as

\displaystyle \lim_{x\to+\infty} h(x)

and

\displaystyle \lim_{x\to-\infty} h(x)

. c. (iii) Determine the coordinates of the points on the graph for which

h(x)=1

and

h(x)=-1

. d. (iv) Sketch the graph of

h

indicating the branch in each quadrant and the behaviour of the curve near

x=3

and for large values of

|x|

. Explain how the negative sign in the numerator affects the placement of the branches compared with

\dfrac{2}{x-3}