91Ó°ÊÓ

Let \(h(x)=\left\\{\begin{array}{ll}x & \text { if } x < 0 \\ x^{2} & \text { if } 0 < x \leq 2 \\ 8-x & \text { if } x > 2\end{array}\right.\) (a) Evaluate each limit, if it exists. (i) \(\lim _{x \rightarrow 0^{+}} h(x)\) (iv) \(\lim _{x \rightarrow 2^{-}} h(x)\) (ii) \(\lim _{x \rightarrow 0} h(x)\) (v) \(\lim _{x \rightarrow 2^{+}} h(x)\) (iii) \(\lim _{x \rightarrow 1} h(x)\) (vi) \(\lim _{x \rightarrow 2} h(x)\) (b) Sketch the graph of \(h\)

(a) What is wrong with the following equation? $$ \frac{x^{2}+x-6}{x-2}=x+3 $$ (b) In view of part (a), explain why the equation $$ \lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x-2}=\lim _{x \rightarrow 2}(x+3) $$ is correct.

In the theory of relativity, the Lorentz contraction formula $$ L=L_{0} \sqrt{1-v^{2} / c^{2}} $$ expresses the length \(L\) of an object as a function of its velocity \(v\) with respect to an observer, where \(L_{0}\) is the length of the object at rest and \(c\) is the speed of light. Find \(\lim _{v \rightarrow c^{-}} L\) and interpret the result. Why is a left-hand limit necessary?

Limits of Sums and Products (a) Show by means of an example that \(\lim _{x \rightarrow a}[f(x)+g(x)]\) may exist even though neither \(\lim _{x \rightarrow a} f(x)\) nor \(\lim _{x \rightarrow a} g(x)\) exists. (b) Show by means of an example that \(\lim _{x \rightarrow a}[f(x) g(x)]\) may exist even though neither \(\lim _{x \rightarrow a} f(x)\) nor \(\lim _{x \rightarrow a} g(x)\) exists.