Exemplos 7

Calcule, caso existam, cada um dos seguintes limites:

1. $\displaystyle \lim_{x \to +\infty} \frac{1}{x-3}$

2. $\displaystyle \lim_{x \to 3} \frac{1}{x-3}$

3. $\displaystyle \lim_{x \to 2^-} \frac{1}{2-x}$

4. $\displaystyle \lim_{x \to -1^+} \frac{1}{x^2-1}$

5. $\displaystyle \lim_{x \to 2} \ln(x-2)$

6. $\displaystyle \lim_{x \to 0} e^{\frac{1}{x}}$

Calcule, caso existam, cada um dos seguintes limites:

1. $\displaystyle \lim_{x \to + \infty} \frac{3^{x} - 2^{x}}{3^{x+1}+ 2^{x-3}}$

2. $\displaystyle \lim_{x \to +\infty} \frac{3x^2-x-10}{x^2-x-2}$

3. $\displaystyle \lim_{x \to 2} \frac{\sqrt{x}-\sqrt{2}}{x-2}$

4. $\displaystyle \lim_{t \to 0} \frac{1-\cos(t)}{\sin(t)}$

5. $\displaystyle \lim_{t \to 2} \frac{e^{2t-4}-1}{t-2}$

6. $\displaystyle \lim_{t \to +\infty}x^2\sin\frac{1}{x}$

7. $\displaystyle \lim_{x \to -\infty}xe^x$

8. $\displaystyle \lim_{x \to +\infty}\left(\sqrt{x}-\sqrt{x+1}\right)$

9. $\displaystyle\lim_{x \to +\infty}\left(\ln(3x^2+2)-\ln (x^2)\right)$

10. $\displaystyle \lim_{x \to \frac{\pi}{2}^+}\frac{\cos(x)}{1-\sin(x)}$

Levantamento de indeterminações do tipo $\infty^0,\;0^0,\;1^\infty$

Calcule, caso existam, cada um dos seguintes limites:

1. $\displaystyle \lim_{x \to 0^+}(1+2x)^{\frac{1}{x}}$ 2. $\displaystyle \lim_{x \to +\infty}x^{\frac{1}{x}}$ 3. $\displaystyle\lim_{x \to 0^+}x^x$