MATH 151. SUMMER 1999.

Review 1

1.

Find limits (if they exist):

(a)

$$\mathop{\lim}_{x\rightarrow 5+} \frac{x^2-7x+10}{x-5}.$$

(b)

$$\mathop{\lim}_{x\rightarrow 5+} \frac{x^2-7x+11}{x-5}.$$

(c)

$$\mathop{\lim}_{x\rightarrow -2} \frac{\sqrt{x+5}-\sqrt{1-x}}{x+2}.$$

(d)

$$\mathop{\lim}_{x\rightarrow 1} (\vert 2x-3\vert-\vert 2x+1\vert).$$

(e)

$$\mathop{\lim}_{x\rightarrow 1} \sin (\cos x).$$

2.

Find the equation of the tangent line to the curve y=x³+2x+1 at the point x=1.

3.

The displacement of an object im a straight line is s=1+t+t². Find the instanteneous velocity at the time t=1.

4.

Using the precise definition of a limit, find

$$\mathop{\lim}_{x\rightarrow 1} (x^2+x).$$

5.

Let

$$f(x)=\left\{ \begin{tabular}{rc} $-x^2,$ & $x\le -1$,\\ $-1,$ & $-1<x\le 0$,\\ $\frac{1}{x},$ & $x>0$. \end{tabular}\right.$$

Sketch the graph of f and find the points of continuity of f.

6.

Show that the equation x³+x+1=0 has at least one solution.