LIMIT AND CONTINUITY

Dz— (z,u,q,a—)

Problems: 1. Give the ϵ − δ definition of lim_{x→a} f(x) = l.

2. Use ϵ − δ definition to prove the following :

if lim_{x→a} f(x) = l, lim_{x→a} g(x) = m

then lim_{x→a} f(x) + g(x) = l + m

3. State the Sandwich Theorem. (i.e. Squeezing Theorem )

4. Give the definition of ”f is continuous at a”.

5. f(x)=\{ \begin{matrix}x^{2} + 2 x < 1 \\ \frac{1}{x+1} x > 1 \end{matrix}

write down (1) lim_{x→1^{−}} f(x) (2) lim_{x→1^{+}} f(x) (3) lim_{x→1} f(x)

6. g(x)\{ \begin{matrix}x^{3} + 1 \\ 3 \\ x + 2 \end{matrix} \begin{matrix}x < 0 \\ x = 0 \\ x > 0 \end{matrix}

(i)write down (1) lim_{x→0^{−}} g(x) (2) lim_{x→0^{+}} g(x) (3) lim_{x→0} g(x) (ii)Is g continuous at 0 ? why ?

7. h(x)=\{ \begin{matrix}x^{2} + 1 \\ 2 \\ −2x \end{matrix} \begin{matrix}x < −1 \\ x = 1 \\ x > −1 \end{matrix}

(i)write down (1) lim_{x→−1^{−}} h(x) (2) lim_{x→−1^{+}} h(x) (3) lim_{x→−1} h(x) (ii)Is h continuous at -1 ? why ?

8. f(x)=\{ \begin{matrix}x \\ −x \end{matrix} \begin{matrix}x is rational \\ x is irrational \end{matrix}

(i)write down (1) lim_{x→0} f(x) (2) lim_{x→1} f(x) (ii)Write down the points at which f is continuous.

2 LIMIT AND CONTINUITY

9. write down lim_{x→0} sin(\frac{1}{x}) , and justify your answer.

10. write down lim_{x→0} x sin(\frac{1}{x}) , and justify your answer.

11. give the definition of lim_{x→a} f(x) = ∞