Find the limits. lim_{xrightarrow-infty}frac{2^x+4^x}{5^x-2^x}

Question
Limits and continuity
Find the limits.
$$\lim_{x\rightarrow-\infty}\frac{2^x+4^x}{5^x-2^x}$$

2020-12-22
Consider the limit
$$\lim_{x\rightarrow-\infty}\frac{2^x+4^x}{5^x-2^x}$$
Solve:
$$\lim_{x\rightarrow-\infty}\frac{2^x+4^x}{5^x-2^x}$$
Take $$2^x$$ common from numerator and denominator
$$=\lim_{x\rightarrow-\infty}\frac{2^x(1+\frac{4^x}{2^x})}{2^x(\frac{5^x}{2^x}-1)}$$
$$=\lim_{x\rightarrow-\infty}\frac{2^x(1+(\frac{4}{2})^x)}{2^x((\frac{5}{2})^x-1)}$$
$$=\lim_{x\rightarrow-\infty}\frac{(1+2^x)}{((2.5)^x-1)}$$
$$=\frac{\lim_{x\rightarrow-\infty}(1+2^x)}{\lim_{x\rightarrow-\infty}((2.5)^x-1)}$$
$$=\frac{(1+2^{-\infty})}{(2.5^{-\infty}-1)}$$
$$=\frac{(1+0)}{(0-1)}$$
$$=-1$$

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