if ( 6x leq g(x) leq 3x^{4}-3x^{2}+6 ) for all ( x ), evaluate ( lim _{x \rightarrow 1} g(x) ).
if ( 6x leq g(x) leq 3x^{4}-3x^{2}+6 ) for all ( x ), evaluate ( lim _{x \rightarrow 1} g(x) ).
Answer
Explanation:
Step1: Find $\lim_{x\rightarrow1}6x$
Substitute (x = 1) into (6x). (\lim_{x\rightarrow1}6x=6\times1 = 6)
Step2: Find $\lim_{x\rightarrow1}(3x^{4}-3x^{2}+6)$
Use the sum - difference rule (\lim_{x\rightarrow a}(u(x)\pm v(x)\pm w(x))=\lim_{x\rightarrow a}u(x)\pm\lim_{x\rightarrow a}v(x)\pm\lim_{x\rightarrow a}w(x)) and the power rule (\lim_{x\rightarrow a}x^{n}=a^{n}). (\lim_{x\rightarrow1}(3x^{4}-3x^{2}+6)=3\lim_{x\rightarrow1}x^{4}-3\lim_{x\rightarrow1}x^{2}+\lim_{x\rightarrow1}6) (=3\times1^{4}-3\times1^{2}+6) (=3 - 3+6=6)
Since (6x\leq g(x)\leq3x^{4}-3x^{2}+6) for all (x), and (\lim_{x\rightarrow1}6x=\lim_{x\rightarrow1}(3x^{4}-3x^{2}+6) = 6)
Answer:
(6)