## Exercises - Limit Laws

1. Evaluate $\displaystyle{\lim_{x \rightarrow \pi/2} \cos^2 (2x)}$ and provide a graphical interpretation

2. Evaluate $\displaystyle{\lim_{x \rightarrow 3} \frac{x-3}{x^2-9}}$ and provide a graphical interpretation

3. Evaluate $\displaystyle{\lim_{x \rightarrow -3} \frac{x-3}{x^2-9}}$ and provide a graphical interpretation

4. Evaluate $\displaystyle{\lim_{x \rightarrow 4} \frac{2 - \sqrt{8-x}}{x-4}}$ and provide a graphical interpretation

5. Given $\displaystyle{f(x) = \left\{ \begin{array}{lll} x + 1 &\textrm{ if }& x \lt 0\\ e^x &\textrm{ if }& x \ge 0 \end{array} \right\}}$

Find the following limits and provide graphical interpretations of each

1. $\displaystyle{\lim_{x \rightarrow 0} f(x)}$

2. $\displaystyle{\lim_{x \rightarrow -1} f(x)}$

6. Evaluate $\displaystyle{\lim_{x \rightarrow 0} \frac{\sin 3x}{x}}$ and provide a graphical interpretation

7. Evaluate $\displaystyle{\lim_{x \rightarrow 0} \frac{1-\cos x}{3x}}$ and provide a graphical interpretation

8. Given $\displaystyle{g(x) = \left\{ \begin{array}{lll} 5x-2 &\textrm{ if }& x \lt 1\\ a &\textrm{ if }& x = 1\\ ax^2 + bx &\textrm{ if }& x \gt 1 \end{array} \right.}$

Determine values for $a$ and $b$ so that $\displaystyle{\lim_{x \rightarrow 1} g(x) = g(1)}$.