Let

$\displaystyle f(x) = \begin{cases} \displaystyle {{1}\over{\left\vert x-1\right\vert }} & \mbox{ if $x \neq 1 $ } \\ 0 & \mbox{ if $x = 1 $ } \end{cases}$

Then answer the following questions.

  1. Find $\displaystyle\lim_{h\to 0} \frac{ f\left(h+1\right)-f\left(1\right)}{h}$ .

    $\displaystyle \infty $ $\displaystyle -\infty $ 0 $\displaystyle$   Does not exist

  2. Find the equation of the tangent line to $y = f(x)$ at $x = {{1}\over{2}} $ .

    y = 4x + 1 y = 4x y = −4x y = 1 − 4x

  3. Find the derivative of $f(x)$ at $x = {{1}\over{2}} $ .

    4 −4 −2 2

  4. Find $f'(a)$ when $a$    < $1 $ .

    $\displaystyle {{1}\over{\left\vert a-1\right\vert }}$ $\displaystyle {{1}\over{a-1}}$ $\displaystyle -{{1}\over{\left(a-1\right)^2}}$ $\displaystyle {{1}\over{\left(a-1\right)^2}}$ $\displaystyle -{{1}\over{a-1}}$

  5. Is $f(x)$ differentiable at $x = 1$ ?

    $\displaystyle$    Yes $\displaystyle$    No


Department of Mathematics
Last modified: 2026-06-24