Basic limit problems


  1. $\lim\limits_{x \to 3}\ x = \ ?$

    Answer:

    $$\lim\limits_{x \to 3}\ x = 3.$$

  2. $\lim\limits_{x \to a} \ \left( x^2 + 7 \right) = \ ?$

    Answer:

    $$\lim\limits_{x \to a} \ \left( x^2 + 7 \right) = a^2 + 7.$$

  3. $\lim\limits_{x \to \pi}\ \cos \left( \dfrac{x}{2} \right) = \ ?$

    Answer:

    \begin{eqnarray} \lim\limits_{x \to \pi}\ \cos \left( \dfrac{x}{2} \right) &=& \cos \left( \dfrac{\pi}{2}\right) \\ \ &=& 1. \end{eqnarray}

  4. $\lim\limits_{x \to \infty}\ e^{-x} =\ ?$

    Answer:

    \begin{eqnarray} \lim\limits_{x \to \infty}\ e^{-x} &=& \ e^{-\infty} \\ \ &=& 1. \end{eqnarray}

  5. $\lim\limits_{x \to a}\ \dfrac{x - 3}{x^2 + 7} = \ ?$

    Answer:

    $$\lim\limits_{x \to a}\ \dfrac{x - 3}{x^2 + 7} = \ \dfrac{a- 3}{a^2 + 7}.$$

  6. $\lim\limits_{x \to \pi}\ x \cos x = \ ?$

    Answer:

    \begin{eqnarray} \lim\limits_{x \to \pi}\ x \cos x &=& \pi \cos \pi \\ \ &=& -\!\pi. \end{eqnarray}