## Exercises1.4Miscellaneous

Solve the following equations:

###### 1.

$\pi ^{x+1}=e\text{.}$

$\displaystyle x=\frac{1-\ln \pi }{\ln \pi }\text{.}$
###### 2.

$2^{3^x}=10\text{.}$

$\displaystyle x=-\frac{\log \log 2}{\log 3}\text{.}$
###### 3.

Find the domain of the function $\displaystyle f(x)=\frac{\ln (\ln (\ln x))}{x-3}+\sin x\text{.}$

$(e,3)\cup (3,\infty )\text{.}$

Solve the following problems:

###### 4.

What is meant by saying that $L$ is the limit of $f(x)$ as $x$ approaches $a\text{?}$

What is meant by saying that the function $f(x)$ is continuous at $x=a\text{?}$
State two properties that a continuous function $f(x)$ can have, either of which guarantees the function is not differentiable at $x=a\text{.}$ Draw an example for each.
A corner or a vertical tangent; $y=|x|\text{;}$ $\displaystyle y=x^{\frac{1}{3}}\text{.}$