Question

Calcule os limites a seguir

Answer 1

A)

$\displaystyle f(x)=\frac{4}{x-6}, a = 6\\\\ \lim_{x \to a^+} f(x) = \lim_{x \to 6^+} \frac{4}{x-6} = \frac{4}{0^+}=+\infty\\\lim_{x \to a^-} f(x) = \lim_{x \to 6^-} \frac{4}{x-6} = \frac{4}{0^-}=-\infty$

B)

$\displaystyle f(x)=\frac{3}{1-x}, a=1\\\\ \lim_{x \to a^+} f(x) = \lim_{x \to 1^+} \frac{3}{1-x} = \frac{3}{0^-} = -\infty \\ \lim_{x \to a^-} f(x) = \lim_{x \to 1^-} \frac{3}{1-x} = \frac{3}{0^-} = +\infty$

C)

$\displaystyle f(x)=\frac{2}{|x-5|}, a=5\\\\\lim_{x \to a^+} f(x) = \lim_{x \to 5^+} \frac{2}{|x-5|} = \frac{2}{0^+}=+\infty\\\lim_{x \to a^-} f(x) = \lim_{x \to 5^-} \frac{2}{|x-5|} = \frac{2}{0^+}=+\infty$

D)

$\displaystyle f(x)=\frac{x+5}{x}, a=0\\\\ \lim_{x \to a^+} f(x) = \lim_{x \to 0^+} \frac{x+5}{x} = \frac{5}{0^+} = +\infty \\ \lim_{x \to a^-} f(x) = \lim_{x \to 0^-} \frac{x+5}{x} = \frac{5}{0^-} = -\infty$

E)

$\displaystyle f(x)=\frac{x}{2-x},a=2\\\\ \lim_{x \to a^+} f(x) = \lim_{x \to 2^+} \frac{x}{2-x} = \frac{2}{0^-}=-\infty \\ \lim_{x \to a^-} f(x) = \lim_{x \to 2^-} \frac{x}{2-x} = \frac{2}{0^+}=+\infty$

F)

$\displaystyle f(x)=\frac{x^2}{x-1},a=1\\\\ \lim_{x \to a^+} f(x) = \lim_{x \to 1^+} \frac{x^2}{x-1} = \frac{1}{0^+}=+\infty\\ \lim_{x \to a^-} f(x) = \lim_{x \to 1^-} \frac{x^2}{x-1} = \frac{1}{0^-}=-\infty$

G)

$\displaystyle f(x)=\frac{1}{x},a=0\\\\\lim_{x \to a^+} f(x) = \lim_{x \to 0^+} \frac{1}{x} = \frac{1}{0^+}=+\infty\\\lim_{x \to a^-} f(x) = \lim_{x \to 0^-} \frac{1}{x} = \frac{1}{0^-}=-\infty$

H)

$\displaystyle f(x)=\frac{1}{x^2},a=0\\\\\lim_{x \to a^+} f(x) = \lim_{x \to 0^+} \frac{1}{x^2} = \frac{1}{0^+}=+\infty\\\lim_{x \to a^-} f(x) = \lim_{x \to 0^-} \frac{1}{x^2} = \frac{1}{0^+}=+\infty$