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Question Number 119253 by bemath last updated on 23/Oct/20

$$Giventhreefunctionf\left(x\right),g\left(x\right)andh\left(x\right).$$$$wheref\left(x\right)=x^2+x-2andh\left(x\right)=x^2+2x-1.$$$$If\frac{f\left(x\right)}{x+3}⩽\frac{g\left(x\right)}{x+3}⩽\frac{h\left(x\right)}{x+3},thenthevalueof$$$$\underset{x\to -1}{\lim }g\left(x\right)=?$$

Answered by benjo_mathlover last updated on 23/Oct/20

$$f\left(x\right)=(x+2)(x-1)$$$$h\left(x\right)={(x+1)}^2-2$$$$\underset{x\to -1}{\lim }\frac{f\left(x\right)}{x+3}⩽\underset{x\to -1}{\lim }\frac{g\left(x\right)}{x+3}⩽\underset{x\to -1}{\lim }\frac{h\left(x\right)}{x+3}$$$$-\frac{1}{2}\underset{x\to -1}{\lim }f\left(x\right)⩽-\frac{1}{2}\underset{x\to -1}{\lim }g\left(x\right)⩽-\frac{1}{2}\underset{x\to -1}{\lim }h\left(x\right)$$$$\iff \underset{x\to -1}{\lim }h\left(x\right)⩽\underset{x\to -1}{\lim }g\left(x\right)⩽\underset{x\to -1}{\lim }f\left(x\right)$$$$where\underset{x\to -1}{\lim }h\left(x\right)=-2\wedge \underset{x\to -1}{\lim }f\left(x\right)=-2$$$$soweget\underset{x\to -1}{\lim }g\left(x\right)=-2$$

Answered by 1549442205PVT last updated on 23/Oct/20

$$\mathrm{when}\mathrm{x}\to -1\mathrm{then}\mathrm{x}+3>0.\mathrm{Hence}$$$$\frac{f\left(x\right)}{x+3}⩽\frac{g\left(x\right)}{x+3}⩽\frac{h\left(x\right)}{x+3}$$$$\iff \mathrm{f}\left(\mathrm{x}\right)⩽\mathrm{g}\left(\mathrm{x}\right)⩽\mathrm{h}\left(\mathrm{x}\right).\mathrm{We}\mathrm{infer}\mathrm{that}$$$$\underset{\to -1}{\lim f}\left(\mathrm{x}\right)⩽\underset{\mathrm{x}\to -1}{\lim g}\left(\mathrm{x}\right)⩽\underset{\mathrm{x}\to -1}{\lim h}\left(\mathrm{x}\right)$$$$\iff \underset{\mathrm{x}\to -1}{\lim }({\mathrm{x}}^2+\mathrm{x}-2)⩽\underset{\mathrm{x}\to -1}{\lim g}\left(\mathrm{x}\right)⩽\underset{\mathrm{x}\to -1}{\lim }({\mathrm{x}}^2+2\mathrm{x}-1)$$$$\iff -2⩽\underset{\mathrm{x}\to -1}{\lim g}\left(\mathrm{x}\right)⩽-2$$$$\Rightarrow \underset{\mathrm{x}\to -1}{\lim g}\left(\mathrm{x}\right)=-2$$

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