Answer to Question #145763 in Calculus for liam donohue

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