Advanced Calculus I -- Fall 2016 -- midterm

Name:_________________________ $\qquad$ 6 problems, 60 points total

• Write the problems in the given order, each problem on a separate page.

$\bullet$ 1. From the definition, prove that $f(x)=x^3$ is continuous at $x=1$.

$\bullet$ 2. From the definition, prove that any subsequence of a convergent sequence converges.

$\bullet$ 3. Is it possible to have a function $f$, a point $a$, and a sequence $\{x_n\}$ convergent to $a$ such that

• $\lim_{x\to a} f(x)$ does not exist but
• $\lim_{n\to \infty} f(x_n)$ does?

$\bullet$ 4. From the definition, prove that l.u.b. of the set $S=\{x=1-1/n: n\in {\bf N}\}$ is $1$.

$\bullet$ 5. From the definition, prove that if $f$ is continuous at $a$ then so is $-f$.

$\bullet$ 6. Suppose $$\lim_{n\to \infty} x_n=\infty \text{ and } \lim_{n\to \infty} y_n=-\infty.$$ What are the possible outcomes for $$\lim_{n\to \infty} (x_n+y_n)?$$ Give an example for each.