1.15: A Rational Curve, Asymptotes and a Tangent

A curve $\displaystyle C$ has equation

(a) Write down an equation of the asymptote to $\displaystyle C$ that is

(i) parallel to the $\displaystyle y$-axis,

(ii) parallel to the $\displaystyle x$-axis.

(b) Find the coordinates of the points of intersection of $\displaystyle C$ with the coordinate axes.

(c) Using calculus show that at every point on the curve, the gradient of $\displaystyle C$ is positive.

(d) Using the axes provided, sketch $\displaystyle C$, showing clearly the asymptotes and the coordinates of the points of intersection of $\displaystyle C$ with the coordinate axes.

The line $\displaystyle l$ is the tangent to $\displaystyle C$ at the point on the curve where $\displaystyle x=1$.

(e) Find an equation of $\displaystyle l$, giving your answer in the form