Let

$\displaystyle f(x) = \begin{cases} \displaystyle {{1}\over{\left\vert x-3\right\vert }} & \mbox{ if $x \neq 3 $ } \\ 0 & \mbox{ if $x = 3 $ } \end{cases}$

Then answer the following questions.

  1. Find $\displaystyle\lim_{h\to 0} \frac{ f\left(h+3\right)-f\left(3\right)}{h}$ .

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

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

    $\displaystyle {{2}\over{3}}$ $\displaystyle {{4}\over{9}}$ $\displaystyle -{{4}\over{9}}$ $\displaystyle -{{2}\over{3}}$

  3. Find the equation of the tangent line to $y = f(x)$ at $x = {{3}\over{2}} $ .

    $\displaystyle y=-{{4\,x}\over{9}}$ $\displaystyle y={{4\,x}\over{9}}+1$ $\displaystyle y=1-{{4\,x}\over{9}}$ $\displaystyle y={{4\,x}\over{9}}$

  4. Find $f'(a)$ when $a$    < $3 $ .

    $\displaystyle {{1}\over{\left\vert a-3\right\vert }}$ $\displaystyle {{1}\over{\left(a-3\right)^2}}$ $\displaystyle {{1}\over{a-3}}$ $\displaystyle -{{1}\over{a-3}}$ $\displaystyle -{{1}\over{\left(a-3\right)^2}}$

  5. Is $f(x)$ differentiable at $x = 3$ ?

    $\displaystyle$    No $\displaystyle$    Yes


Department of Mathematics
Last modified: 2024-09-15