함수의 극한과 연속 전체 문제

In Exercises 11–18, a graph of a function $f$ is given along with a value $a$. Determine if $f$ is continuous at $a$; if it is not, state why it is not.

(a) $a = -2$ (b) $a=0$ (c) $a=2$

In Exercises 11–18, a graph of a function $f$ is given along with a value $a$. Determine if $f$ is continuous at $a$; if it is not, state why it is not.

$a = 3\pi/2$

In Exercises 19–22, determine if $f$ is continuous at the indicated values. If not, explain why.

$\displaystyle f(x) = \left\{\begin{array}{ccc} 1 & & x=0\\ \frac{\sin x}{x} & & x>0 \end{array}\right.$

(a) $x=0$ (b) $x=\pi$

In Exercises 19–22, determine if $f$ is continuous at the indicated values. If not, explain why.

$\displaystyle f(x) = \left\{\begin{array}{ccc} x^3-x & & x<1\\ x-2 & & x\geq 1 \end{array}\right.$

(a) $x=0$ (b) $x=1$

In Exercises 19–22, determine if $f$ is continuous at the indicated values. If not, explain why.

$\displaystyle f(x) = \left\{\begin{array}{ccc} \frac{x^2+5x+4}{x^2+3x+2} & & x\neq -1\\ 3 & & x=-1 \end{array}\right.$

(a) $x=-1$ (b) $x=10$

In Exercises 19–22, determine if $f$ is continuous at the indicated values. If not, explain why.

$\displaystyle f(x) = \left\{\begin{array}{ccc} \frac{x^2-64}{x^2-11 x+24} & & x\neq 8\\ 5 & & x=8 \end{array}\right.$

(a) $x=0$ (b) $x=8$

In Exercises 23–34, give the intervals on which the given function is continuous.

$f(x) = x^2-3x+9$

In Exercises 23–34, give the intervals on which the given function is continuous.

$\displaystyle g(x) = \sqrt{x^2-4}$

In Exercises 23–34, give the intervals on which the given function is continuous.

$\displaystyle g(x) = \sqrt{4-x^2}$

In Exercises 23–34, give the intervals on which the given function is continuous.

$\displaystyle h(k) = \sqrt{1-k}+\sqrt{k+1}$

In Exercises 23–34, give the intervals on which the given function is continuous.

$\displaystyle f(t) = \sqrt{5t^2-30}$

In Exercises 23–34, give the intervals on which the given function is continuous.

$\displaystyle g(t) = \frac{1}{\sqrt{1-t^2}}$

In Exercises 23–34, give the intervals on which the given function is continuous.

$\displaystyle g(x) = \frac{1}{1+x^2}$

In Exercises 23–34, give the intervals on which the given function is continuous.

$\displaystyle f(x) = e^x$

In Exercises 23–34, give the intervals on which the given function is continuous.

$\displaystyle g(s) = \ln s$

In Exercises 23–34, give the intervals on which the given function is continuous.

$\displaystyle h(t) = \cos t$

In Exercises 23–34, give the intervals on which the given function is continuous.

$\displaystyle f(k) = \sqrt{1-e^k}$

In Exercises 23–34, give the intervals on which the given function is continuous.

$\displaystyle f(x) = \sin(e^x+x^2)$

Exercises 35–38 test your understanding of the Intermediate Value Theorem.

Let $f$ be continuous on $[1,5]$ where $f(1) = -2$ and $f(5) = -10$. Does a value $1<c<5$ exist such that $f(c) = -9$? Why/why not?

Exercises 35–38 test your understanding of the Intermediate Value Theorem.

Let $g$ be continuous on $[-3,7]$ where $g(0) = 0$ and $g(2) = 25$. Does a value $-3<c<7$ exist such that $g(c) = 15$? Why/why not?