함수의 극한과 연속 전체 문제

In Exercises 13–21, evaluate the given limits of the piecewise defined functions $f$.

$\displaystyle f(x) = \left\{\begin{array}{ccc} x+1 & & x<1 \\ 1 & & x=1\\ x-1 & & x>1 \end{array} \right.$

(a) $\displaystyle \lim_{x\to 1^-} f(x)$ (b) $\displaystyle \lim_{x\to 1^+} f(x)$

(a) $\displaystyle \lim_{x\to 1} f(x)$ (b) $f(1)$

In Exercises 13–21, evaluate the given limits of the piecewise defined functions $f$.

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

(a) $\displaystyle \lim_{x\to 2^-} f(x)$ (b) $\displaystyle \lim_{x\to 2^+} f(x)$

(a) $\displaystyle \lim_{x\to 2} f(x)$ (b) $f(2)$

In Exercises 13–21, evaluate the given limits of the piecewise defined functions $f$.

$\displaystyle f(x) = \left\{\begin{array}{ccc} a(x-b)^2+c & & x<b \\ a(x-b)+c & & x\geq b \end{array} \right.,$

where $a$, $b$ and $c$ are real numbers.

(a) $\displaystyle \lim_{x\to b^-} f(x)$ (b) $\displaystyle \lim_{x\to b^+} f(x)$

(a) $\displaystyle \lim_{x\to b} f(x)$ (b) $f(b)$

In Exercises 13–21, evaluate the given limits of the piecewise defined functions $f$.

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

(a) $\displaystyle \lim_{x\to 0^-} f(x)$ (b) $\displaystyle \lim_{x\to 0^+} f(x)$

(a) $\displaystyle \lim_{x\to 0} f(x)$ (b) $f(0)$

In your own words, describe what it means for a function to be continuous.

In your own words, describe what the Intermediate Value Theorem states.

What is a “root” of a function?

Given functions $f$ and $g$ on an interval $I$, how can the Bisection Method be used to find a value $c$ where $f(c) = g(c)$?

T/F: If $f$ is defined on an open interval containing $c$, and $\displaystyle \lim_{x\to c}f(x)$ exists, then $f$ is continuous at $c$.

T/F: If $f$ is continuous at $c$, then $\displaystyle \lim_{x\to c}f(x)$ exists.

T/F: If $f$ is continuous at $c$, then $\displaystyle \lim_{x\to c^+}f(x) = f(c)$.

T/F: If $f$ is continuous on $[a,b]$, then $\displaystyle \lim_{x\to a^-}f(x) = f(a)$.

T/F: If $f$ is continuous on $[0,1)$ and $[1,2)$, then $f$ is continuous on $[0,2)$.

T/F: The sum of continuous functions is also continuous.

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.

$a = 1$

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.

$a = 1$

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.

$a = 1$

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.

$a = 0$

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.

$a = 1$

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.

$a = 4$