Question

3 a,b,c,d. Calcularea limitelor

Answer 1

Răspuns:

a) Se folosește limita remarcabilă $\displaystyle\lim_{x\to 0}\dfrac{a^x-1}{x}=\ln a$

$\displaystyle\lim_{x\to 0}\dfrac{4^x-1}{3^x-1}=\lim_{x\to 0}\dfrac{\frac{4^x-1}{x}}{\frac{3^x-1}{x}}=\dfrac{\ln 4}{\ln 3}$

b)

$\displaystyle\lim_{x\to 0}\dfrac{\frac{4^x-1}{x}-\frac{3^x-1}{x}}{\frac{3^x-1}{x}-\frac{2^x-1}{x}}=\dfrac{\ln 4-\ln 3}{\ln 3-\ln 2}=\dfrac{\ln\frac{4}{3}}{\ln\frac{3}{2}}$

c) Se folosește limita remarcabilă $\displaystyle\lim_{u(x)\to 0}\dfrac{\ln(1+u(x))}{u(x)}=1$

$\displaystyle\lim_{x\to 0}\dfrac{\ln(1+2x^2}{2x^2}\cdot\dfrac{3x^2}{\ln(1+3x^2)}}\cdot\dfrac{2}{3}=\dfrac{2}{3}$

d)

$\displaystyle\lim_{x\to\infty}(\ln x^2-\ln(x+1))=\lim_{x\to\infty}\ln\dfrac{x^2}{x+1}=\infty$

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