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The function $f$ in the figure satisfies $\lim _{x \rightarrow 2} f(x)=4 .$ Determine the largest value of $\delta>0$ satisfying each statement. a. If $0<|x-2...

Question

The function $f$ in the figure satisfies $\lim _{x \rightarrow 2} f(x)=4 .$ Determine the largest value of $\delta>0$ satisfying each statement. a. If $0<|x-2|<\delta$ then $|f(x)-4|<1$ b. If $0<|x-2|<\delta$ then $|f(x)-4|<1 / 2$

The function $f$ in the figure satisfies $\lim _{x \rightarrow 2} f(x)=4 .$ Determine the largest value of $\delta>0$ satisfying each statement. a. If $0<|x-2|<\delta$ then $|f(x)-4|<1$ b. If $0<|x-2|<\delta$ then $|f(x)-4|<1 / 2$



Answers

Using the Definition of Limits at Infinity The graph of
$$f(x)=\frac{2 x^{2}}{x^{2}+2}$$
is shown (see figure).
(a) Find $$L=\lim _{x \rightarrow \infty} f(x)$$
(b) Determine $x_{1}$ and $x_{2}$ in terms of $\varepsilon$
(c) Determine $M,$ where $M>0,$ such that $|f(x)-L|<\varepsilon$ for
$x>M .$
(d) Determine $N,$ where $N<0,$ such that $|f(x)-L|<\varepsilon$ for $x<N .$


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