Function $f(x)$ is given as $f(x) = x + \dfrac{1}{x}$. I'm asked to find the line of a tangent and normal at $x = 1$.
I'm able to obtain the gradient, m, of the function at $ x = 1$. Since the first derivative of the $f(x)$ is $f'(x) = 1 - \dfrac{1}{x^2}$, then
$$m_t = f'(1) = 0.$$
Consequently, the equation of the tangent line is:
$$y-2 = m_t (x - 1) \implies y = 2$$
But, using the properity that $m_{n} \times m_{t} = -1 $, we get that the equation of a normal line is undefined since $$m_{n} = -\dfrac{1}{0}.$$
My question here is, is the correct answer to say that the equation of the normal is not defined or something else?
This means Eq. of tangent is $y=2$ and the normal is $y-2=m_n(x-1) \implies (x-1)=\frac{y-2}{m_n}=0$. Hence the equation of required normal is $x=1$.