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Question Number 42933 by Joel578 last updated on 05/Sep/18

$$\mathrm{Suppose}\mathrm{that}f\mathrm{and}g\mathrm{are}\mathrm{two}\mathrm{functions}\mathrm{such}\mathrm{that}$$$$\underset{x\to a}{\lim }g\left(x\right)=0\mathrm{and}\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}\mathrm{exist}.$$$$\mathrm{Prove}\mathrm{that}\underset{x\to a}{\lim }f\left(x\right)=0$$

Commented by MrW3 last updated on 05/Sep/18

$$h\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$$$$\Rightarrow f\left(x\right)=h\left(x\right)g\left(x\right)$$$$\underset{x\to a}{\lim }f\left(x\right)=\underset{x\to a}{\lim }h\left(x\right)\times \underset{x\to a}{\lim }g\left(x\right)=b\times 0=0$$

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