One-sided limit problems

  1. Let $$f (x) = \left\lbrace \begin{matrix} x + 2, & \text{if $x < 0$} \\ 3x-7, & \text{if $x \ge 0$} \end{matrix} \right. ,$$ then $$\lim\limits_{x \to 0^+}\ f(x)= \ ?$$

    Answer:

    \begin{eqnarray*} \lim\limits_{x \to 0^+}\ f(x) &=& \lim\limits_{x \to 0^+} \left(3x-7\right) \\ \ &=& 0 - 7 \\ \ &=& 7. \end{eqnarray*}
  2. Let $$f (x) = \left\lbrace \begin{matrix} x + 2,\ \text{if $x < 0$} \\ 3x-7 ,\ \text{if $x \ge 0$} \end{matrix} \right.$$ then $$\lim\limits_{x \to 0^-}\ f(x)= \ ?$$

    Answer:

    \begin{eqnarray*} \lim\limits_{x \to 0^-}\ f(x) &=& \lim\limits_{x \to 0^-} \left(x + 2 \right) \\ \ &=& 0 + 2 \\ \ &=& 2. \end{eqnarray*}
  3. Let $$f (x) = \left\lbrace \begin{matrix} x + 2, & \text{if $x < 0$} \\ 3x-7, & \text{if $x \ge 0$} \end{matrix} \right. ,$$ then $$\lim\limits_{x \to 0}\ f(x)= \ ?$$

    Answer:

    $$\lim\limits_{x \to 0}\ f(x) \text{ does not exist}$$ since$$\lim\limits_{x \to 0^+}\ f(x) \neq \lim\limits_{x \to 0^-}\ f(x).$$ Recall from the previous two questions that $$\lim\limits_{x \to 0^-}\ f(x)= 2$$ and $$\lim\limits_{x \to 0^+}\ f(x)= -7.$$