Question

# 1. Let f be a function defined for all x, and continuous at all values of x except at –2 and 0. Suppose that \text { for all } x>10 \text { we have that } b \leq f(x) \leq \frac{b x^{2}+x}{\gamma^{2}-b} . \text { Moreover, suppose that the following is known: } \lim _{x \rightarrow-2^{-}} f(x)=3, \quad \lim _{x \rightarrow-2^{+}} f(x)=f(-2)=1, \quad \lim _{x \rightarrow 0^{-}} f(x)=f(0)=-b, \quad \lim _{x \rightarrow 0^{+}} f(x)=2 (a) Does there always exist a value c on the interval (–3,0) such that f(c)0? Justify your answer. (b) Does there always exist a value c on the interval (–1,1) such that f(c) = 0? Justify your answer. (c) Evaluate the following limit: \lim _{x \rightarrow \infty} f(x) Hint: In (a) and (b) we want to know if there always exists such a value c. Finding a specific example where such a c exists does not show that something always happens. The only thing you are allowed to assume about f(x) is the information given in the prompt.  Fig: 1  Fig: 2  Fig: 3  Fig: 4  Fig: 5  Fig: 6  Fig: 7  Fig: 8