Question

Evaluate:
$\rm \displaystyle \lim_{x \to 0}\ \frac{49^{x}-2(35^{x})+25^{x}}{x^{2}}$

Answer 1

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Answer 2

we have to evaluate, $\rm \displaystyle \lim_{x \to 0}\ \frac{49^{x}-2(35^{x})+25^{x}}{x^{2}}$

putting, x = 0

we get “ 0/0 ” is the form of limit.

so, $\rm \displaystyle \lim_{x \to 0}\ \frac{49^{x}-2(35^{x})+25^{x}}{x^{2}}$

= $\displaystyle\lim_{x\to 0}\frac{(7^x)^2-2(7^x)(5^x)+(5^x)^2}{x^2}$

= $\displaystyle\lim_{x\to 0}\frac{(7^x-5^x)^2}{x^2}$

= $\displaystyle\lim_{x\to 0}\left(\frac{7^x-5^x}{x}\right)^2$

= $\left(\displaystyle\lim_{x\to 0}\frac{(7^x-1)-(5^x-1)}{x}\right)^2$

we know, $\displaystyle\lim_{x\to 0}\frac{a^x-1}{x}=loga$

= $\left(\displaystyle\lim_{x\to 0}\frac{\frac{(7^x-1)}{x}\times x-\frac{(5^x-1)}{x}\times x}{x}\right)^2$

= [log7 - log5]²

= [log(7/5)]²

= log²(7/5) [ans]