Section Assignments # 3 & #4- Due Tuesday 2/11
- (Some of you finished this in class. If you did, just move on to the next problem.)
Consider the function \(f:[-1,1]\rightarrow \mathbb{R}\) by
\[
f(x)=
\begin{cases}
1+2x^2, & x\; \textrm{rational}\\
1+x^4, & x\; \textrm{irrational}.
\end{cases}
\]
(Recall a rational number is a number that can be written as a quotient of integers.)
Find \(\lim_{x\rightarrow 0} f(x)\) and clearly justify your answer.
Hint: Can you find both lower and upper bounds for \(f(x)\)? Your bounds should hold for any \(x\) in the domain of \(f\).
-
Consider the functions \(f(x)=1\) and \(g(x)=\begin{cases}2x, & x \neq 1\\ 5, & x=1. \end{cases}\)
- Calculate \(\lim_{x \rightarrow 0} f(x)\).
- Calculate \(\lim_{x \rightarrow 1} g(x)\).
- Find an expression for the composition \(g\circ f (x)=g(f(x))\).
- Using your answer to part c, calculate \(\lim_{x \rightarrow 0} g(f(x))\).
- Based on your above answers, do you think if we have two functions \(h(x)\) and \(j(x)\) which statisfy \(\lim_{x\rightarrow c}h(x)=L\;\) and \(\;\lim_{x\rightarrow L}j(x)=M,\;\) then it is safe to say that \(\;\lim_{x\rightarrow c}j(h(x))=M\,\)?