Question

Shown below are six statements about functions. Match each statement to one of the functions shown below which BEST matches that statement.

1. limx→6+f(x) and limx→6−f(x) both exist and are finite, but they are not equal.
2. The graph of y=f(x) has vertical tangent line at (6,f(6))
3. limx→6−f(x)=−∞.
4. limx→6+f(x) exists but limx→6−f(x) does not.
5. limx→6f(x)=∞.
6. limx→6f(x) exists but f is not continuous at 6.

A. f(x)=⎧⎩⎨⎪⎪⎪⎪cos(1x−6)02x+24if x<6if x=6if x>6
B. f(x)=⎧⎩⎨⎪⎪2x024−2xif x<6if x=6if x>6
C. f(x)=⎧⎩⎨⎪⎪2x02x−24if x<6if x=6if x>6
D. f(x)=x−6−−−−−√3
E. f(x)=1(x−6)2
F. f(x)=1x−6

Answer 1

1. $\lim\limits_{x\to 6^+}f(x)$ and $\lim\limits_{x\to 6^-}f(x)$ both exist and are finite, but they are not equal.

C. $f(x)=x = \begin{cases} 2x &\text{if } x<6 \\ 0 &\text{if } x=6 \\ 2x-24 &\text{if } x>6 \end{cases}$

2. The graph of $y=f(x)$ has vertical tangent line at $(6, f(6))$

D. $f(x)=\sqrt[3]{x-6}$

3.$\lim\limits_{x\to 6^-}f(x)=-\infin$

F. $f(x)=\dfrac{1}{x-6}$

4.$\lim\limits_{x\to 6^+}f(x)$ exists but $\lim\limits_{x\to 6^-}f(x)$ does not

A. $f(x)=x = \begin{cases} \cos(\dfrac{1}{x-6}) &\text{if } x<6 \\ 0 &\text{if } x=6 \\ 2x+24 &\text{if } x>6 \end{cases}$

5.$\lim\limits_{x\to 6}f(x)=\infin$

E. $f(x)=\dfrac{1}{(x-6)^2}$

6. $\lim\limits_{x\to 6}f(x)$ exists but $f$ is not continuous at $6$

B. $f(x)=x = \begin{cases} 2x &\text{if } x<6 \\ 0 &\text{if } x=6 \\ 24-2x &\text{if } x>6 \end{cases}$