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Question Number 181014 by Mastermind last updated on 20/Nov/22

$$\mathrm{If}\mathrm{f}\left(\mathrm{x}\right)=3-{\mathrm{x}}^2,\mathrm{find}\frac{\mathrm{f}(3+\mathrm{h})-\mathrm{f}\left(3\right)}{\mathrm{h}}$$$$$$$$.$$

Commented by CElcedricjunior last updated on 20/Nov/22

$$\mathit{f}\left(\mathit{x}\right)=3-{\mathit{x}}^2$$$$\underset{\mathit{h}\to 0}{\lim }\frac{\mathit{f}(3+\mathit{h})-\mathit{f}\left(3\right)}{\mathit{h}}=\underset{\mathit{h}\to 0}{\lim }\frac{3-9-6\mathit{h}-{\mathit{h}}^2-6}{\mathit{h}}$$$$=\underset{\mathit{h}\to 0}{\lim }\frac{-(12+6\mathit{h}+{\mathit{h}}^2)}{\mathit{h}}=\frac{-12}{0^{\mp }}=\{\begin{array}{l}+\mathrm{\infty }\mathit{s}\mathit{i}{0}^{-}\\ -\mathrm{\infty }\mathit{s}\mathit{i}{0}^{+}\end{array}$$

Commented by mr W last updated on 20/Nov/22

$$maybeyouwanttoask$$$$\mathrm{find}\underset{h\to 0}{\lim }\frac{\mathrm{f}(3+\mathrm{h})-\mathrm{f}\left(3\right)}{\mathrm{h}}?$$

Commented by Mastermind last updated on 20/Nov/22

$$\mathrm{i}\mathrm{think}\mathrm{it}\mathrm{should}\mathrm{be}\mathrm{like}\mathrm{that}$$

Commented by mr W last updated on 20/Nov/22

$$thisisyourquestion.youshouldknow$$$$whatyouwanttoask,ornot?$$

Commented by Frix last updated on 20/Nov/22

$$\frac{f(3+h)-f\left(3\right)}{h}=\frac{3-{(3+h)}^2-(3-3^2)}{h}=$$$$=\frac{3-9-6h-h^2-3+9}{h}=-\frac{6h+h^2}{h}=-6-h$$$$\Rightarrow$$$$\underset{h\to 0}{\lim }\frac{f(3+h)-f\left(3\right)}{h}=f^{\prime }\left(3\right)=-6\mathrm{as}\mathrm{it}\mathrm{should}\mathrm{be}$$

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