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

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

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

    $\displaystyle$    No $\displaystyle$    Yes

  3. Find $f'(a)$ when $a$    > $1 $ .

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

  4. Find the derivative of $f(x)$ at $x = 4 $ .

    $\displaystyle -{{1}\over{9}}$ $\displaystyle -{{1}\over{3}}$ $\displaystyle {{1}\over{3}}$ $\displaystyle {{1}\over{9}}$

  5. Find the equation of the tangent line to $y = f(x)$ at $x = 4 $ .

    $\displaystyle y={{x}\over{9}}+{{7}\over{9}}$ $\displaystyle y={{x}\over{9}}$ $\displaystyle y=-{{x}\over{9}}$ $\displaystyle y={{7}\over{9}}-{{x}\over{9}}$


Department of Mathematics
Last modified: 2026-04-02