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Question Number 174407 by infinityaction last updated on 31/Jul/22

$$evaluate$$$$\underset{x\to -\mathrm{\infty }}{\lim }\frac{3^{\sin x}+2x+1}{\sin x-\sqrt{x^2+1}}$$

Answered by CElcedricjunior last updated on 31/Jul/22

$$\underset{x\to -\mathrm{\infty }}{\lim }\frac{3^{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}+2\mathit{x}+1}{\mathit{s}\mathit{i}\mathit{n}\mathit{x}-\sqrt{{\mathit{x}}^2+1}}$$$$\mathit{o}\mathit{r}\mathrm{\forall }\mathit{x}\in \mathbb{R}-1⩽\mathit{s}\mathit{i}\mathit{n}\mathit{x}⩽1$$$$\iff \frac{1}{3}⩽3^{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}⩽3$$$$\iff \frac{1}{3}+2\mathit{x}+1⩽3^{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}+2\mathit{x}+1⩽4+2\mathit{x}$$$$-1⩽\mathit{s}\mathit{i}\mathit{n}\mathit{x}⩽1$$$$-1-\sqrt{{\mathit{x}}^2+1}⩽\mathit{s}\mathit{i}\mathit{n}\mathit{x}-\sqrt{1+{\mathit{x}}^2}⩽1-\sqrt{1+{\mathit{x}}^2}$$$$\iff \frac{\frac{4}{3}+2\mathit{x}}{-1-\sqrt{1+{\mathit{x}}^2}}⩽\frac{3^{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}+2\mathit{x}+1}{\mathit{s}\mathit{i}\mathit{n}\mathit{x}-\sqrt{1+{\mathit{x}}^2}}⩽\frac{4+2\mathit{x}}{1-\sqrt{1+{\mathit{x}}^2}}$$$$\underset{x\to -\mathrm{\infty }}{\lim }\frac{\frac{4}{3}+2\mathit{x}}{-1-\sqrt{1+{\mathit{x}}^2}}=\underset{x\to -\mathrm{\infty }}{\lim }\frac{\frac{4}{3}+2x}{-1-\mid \mathit{x}\mid \sqrt{1+\frac{1}{{\mathit{x}}^2}}}$$$$=\underset{x\to -\mathrm{\infty }}{\lim }\frac{\mathit{x}(\frac{4}{3\mathit{x}}+2)}{\mathit{x}(-\frac{1}{\mathit{x}}+\sqrt{1+\frac{1}{{\mathit{x}}^2}})}\mathit{c}\mathit{a}\mathit{s}\mathit{q}\mathit{d}:\{\begin{array}{l}\mathit{x}->-\mathrm{\infty }\\ \mid \mathit{x}\mid =-\mathit{x}\end{array}$$$$=2$$$$\underset{x\to -\mathrm{\infty }}{\lim }\frac{4+2\mathit{x}}{1-\sqrt{1+{\mathit{x}}^2}}=2\mathit{e}\mathit{n}\mathit{p}\mathit{r}\mathit{o}\mathit{c}\mathit{e}\mathit{d}\mathit{a}\mathit{n}\mathit{t}\mathit{d}\mathit{e}$$$$\mathit{l}\mathit{a}\mathit{m}\mathit{e}\mathit{m}\mathit{e}\mathit{m}\mathit{a}\mathit{n}\mathit{i}\mathit{e}\mathit{r}\mathit{e}$$$${\mathit{d}}^{\prime }\mathbf{o}\stackrel{`}{\mathbf{u}}\underset{x\to -\mathrm{\infty }}{\lim }\frac{3^{\mathit{s}\mathit{i}\mathit{n}\mathit{x}}+2\mathit{x}+1}{\mathit{s}\mathit{i}\mathit{n}\mathit{x}-\sqrt{1+{\mathit{x}}^2}}=2$$$$\mathcal{D}{}^{\prime }apr\stackrel{`}{es}leth\acute{eor}\stackrel{`}{eme}desgendarmes$$$$$$$$.........lec\acute{el}\stackrel{`}{ebre}cedricjunior...........$$$$$$$$$$

Commented by infinityaction last updated on 01/Aug/22

$$thanks$$

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