(Solved): 2. Find the limits. (a) limx1x22x32x2+3x+1 (b) limt32t2+7t+3t29 (c ...

2. Find the limits. (a) ${\lim }_{x??1}\frac{2x^2+3x+1}{x^2?2x?3}$ (b) ${\lim }_{t??3}\frac{t^2?9}{2t^2+7t+3}$ (c) ${\lim }_{x?16}\frac{4?\sqrt{x}}{16x?x^2}$ (d) ${\lim }_{x?0}\left(x^2?1\right)(2?\cos x)$ (c) ${\lim }_{t?0}\left(\frac{1}{t\sqrt{1+t}}?\frac{1}{t}\right)$ (f) ${\lim }_{h?0}\frac{\frac{1}{(x+h{)}^2}?\frac{1}{x^2}}{h}$ 3. Use the Squeeze Theorem to show that ${\lim }_{x?0}\sqrt{x^3+x^2}\sin \frac{?}{x}=0$ 4. If $f(x)=\frac{2x?1}{?2x^3?x^2?}$, determine whether ${\lim }_{x?0.5}f(x)$ exists. 5. Explain why the function is discontinuous at the given number $a$, and determine which kind : removable or infinite discontinuity or jump discontinuity ? (a) $f(x)=\frac{x^4?1}{x?1},a=1$ (b) $f(x)=\{\begin{array}{cl}1?x^2& \text{ if }x<1\\ \frac{1}{x}& \text{ if }x?1\end{array},a=1$ 6. For what values of the constants $a$ and $b$ are the function $f$ continuous on $(??,?)$ $f(x)=?\begin{array}{cl}\frac{x^2?4}{x?2}& \text{ if }x<2\\ a{x}^2?bx+3& \text{ if }2?x<3\\ 2x?a+b& \text{ if }x?3\end{array}$ 7. Find numbers $a$ and $b$ such that ${\lim }_{x?0}\frac{\sqrt{ax+b}?2}{x}=1$ 8. Show that there is a root of the equation $\cos x=x^3$ between 0 and 1 .