Limit of function

Exercise

1.

Find the limit of the followings:．

(a)

$${\lim_{x \rightarrow 2}(x^2 + 4x)}$$

(b)

$${\lim_{x \rightarrow 2}\frac{x^2 - 3x + 2}{x^2 -4}}$$

(c)

$${\lim_{x \rightarrow 0}\frac{2x^4 - 6x^3 + x^2 + 2}{x - 1}}$$

(d)

$${\lim_{x \rightarrow 1}\frac{2x^4 - 6x^3 + x^2 + 3}{x - 1}}$$

(e)

$${\lim_{x \rightarrow 0}\frac{\sqrt{1+x} - \sqrt{1 - x}}{x}}$$

(f)

$${\lim_{x \rightarrow 0}\frac{(1+x)^{1/3} - (1 - x)^{1/3}}{x}}$$

(g)

$${\lim_{x \rightarrow 2} f(x)}$$, where $${f(x) = \left\{\begin{array}{cl} x^2, & x \neq 2\\ 3, & x = 2 \end{array} \right.}$$

2.

Find the limit of the followings using $${\lim_{x \rightarrow 0}\frac{\sin{x}}{x} = 1}$$

(a)

$${\lim_{x \rightarrow 0}\frac{\sin{2x}}{\sin{3x}}}$$

(b)

$${\lim_{x \rightarrow 0}\frac{\cos{x} - 1}{x}}$$

(c)

$${\lim_{x \rightarrow 0}\frac{\sin^{-1}{x}}{x}}$$

(d)

$${\lim_{x \rightarrow 0}x \sin\frac{1}{x}}$$

3.

Show that if $\lim_{x \rightarrow a}\vert f(x)\vert = 0$，then $\lim_{x \rightarrow a}f(x) = 0$．