Exercises 1.4 Miscellaneous
Solve the following equations:
3.
Find the domain of the function \(\displaystyle f(x)=\frac{\ln (\ln (\ln x))}{x-3}+\sin x\text{.}\)
Answer
\((e,3)\cup (3,\infty )\text{.}\)
Solve the following problems:
4.
What is meant by saying that \(L\) is the limit of \(f(x)\) as \(x\) approaches \(a\text{?}\)
Answer
Give a definition of the limit.
5.
What is meant by saying that the function \(f(x)\) is continuous at \(x=a\text{?}\)
Answer
Give a definition of a function continuous at a point.
6.
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.
Answer
A corner or a vertical tangent; \(y=|x|\text{;}\) \(\displaystyle y=x^{\frac{1}{3}}\text{.}\)