songoku
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- Homework Statement
- Prove that f(x) is continuous at x = 2 where
##f(x)=\begin{cases} x^3 +2, & x \lt 2 \\ 10, & x =2 \\ x^2+6, & x > 2 \end{cases}##
- Relevant Equations
- ##\lim_{x \to a^{-}} f(x)=L## if for every ##\epsilon>0## there is ##\delta >0## such that if ##a-\delta<x<a## then ##|f(x)-L|<\epsilon##
##\lim_{x \to a^{+}} f(x)=L## if for every ##\epsilon>0## there is ##\delta >0## such that if ##a<x<a+\delta## then ##|f(x)-L|<\epsilon##
1) proofing ##\lim_{x \to 2^{-} f(x)=10}##
$$|x^3 +2-10|<\epsilon$$
$$-\epsilon<x^3-8<\epsilon$$
$$8-\epsilon<x^3<8+\epsilon$$
$$\sqrt[3] {8-\epsilon}<x<\sqrt[3] {8+\epsilon}$$
$$\sqrt[3] {8-\epsilon}-2<x-2<\sqrt[3] {8+\epsilon}-2$$
$$|\sqrt[3] {8-\epsilon}-2|<|x-2|<|\sqrt[3] {8+\epsilon}-2|$$
I want to take the minimum of ##{|\sqrt[3] {8-\epsilon}-2|, |\sqrt[3] {8+\epsilon}-2|}## as ##\delta## but how to know which one without using calculator?
2) proofing ##\lim_{x \to 2^{+} f(x)=10}##
$$|x^2+6-10|<\epsilon$$
$$-\epsilon<x^2-4<\epsilon$$
$$4-\epsilon<x^2<4+\epsilon$$
$$\sqrt{4-\epsilon}<x<\sqrt{4+\epsilon}$$
$$\sqrt{4-\epsilon}-2<x-2<\sqrt{4+\epsilon}-2$$
$$|\sqrt{4-\epsilon}-2|<|x-2|<|\sqrt{4+\epsilon}-2|$$
Same thing, how to know the minimum of ##{|\sqrt{4-\epsilon}-2| , |\sqrt{4+\epsilon}-2|}## without calculator?
Thanks
$$|x^3 +2-10|<\epsilon$$
$$-\epsilon<x^3-8<\epsilon$$
$$8-\epsilon<x^3<8+\epsilon$$
$$\sqrt[3] {8-\epsilon}<x<\sqrt[3] {8+\epsilon}$$
$$\sqrt[3] {8-\epsilon}-2<x-2<\sqrt[3] {8+\epsilon}-2$$
$$|\sqrt[3] {8-\epsilon}-2|<|x-2|<|\sqrt[3] {8+\epsilon}-2|$$
I want to take the minimum of ##{|\sqrt[3] {8-\epsilon}-2|, |\sqrt[3] {8+\epsilon}-2|}## as ##\delta## but how to know which one without using calculator?
2) proofing ##\lim_{x \to 2^{+} f(x)=10}##
$$|x^2+6-10|<\epsilon$$
$$-\epsilon<x^2-4<\epsilon$$
$$4-\epsilon<x^2<4+\epsilon$$
$$\sqrt{4-\epsilon}<x<\sqrt{4+\epsilon}$$
$$\sqrt{4-\epsilon}-2<x-2<\sqrt{4+\epsilon}-2$$
$$|\sqrt{4-\epsilon}-2|<|x-2|<|\sqrt{4+\epsilon}-2|$$
Same thing, how to know the minimum of ##{|\sqrt{4-\epsilon}-2| , |\sqrt{4+\epsilon}-2|}## without calculator?
Thanks
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