granice

[malineczka8888]

oblicz:
a) \(\lim_{x \to 5}\) \(\frac{2x^2-11x+5}{3x^2-14x-5}\)
b) \(\lim_{x \to 0}\) \(\frac{1- \sqrt{x+1} }{x}\)
c) \(\lim_{x \to O^-}\) \(\frac{x+1}{1+e^{ \frac{1}{x}} }\)
d) \(\lim_{x \to O^+}\) \(\frac{x+1}{1+e^{ \frac{1}{x}} }\)

[hiohiohio55]

oblicz:
a) \(\lim_{x \to 5}\) \(\frac{2x^2-11x+5}{3x^2-14x-5}\)
b) \(\lim_{x \to 0}\) \(\frac{1- \sqrt{x+1} }{x}\)

a)\(\lim_{x \to 5}\) \(\frac{2x^2-11x+5}{3x^2-14x-5}\)=\(\lim_{x \to 5} \frac{2(x- \frac{1}{2})(x-5) }{3(x+ \frac{1}{3})(x-5) }\)=\(\lim_{x \to 5} \frac{2(x- \frac{1}{2)}}{3(x+ \frac{1}{3}) }\) i teraz za x podstawić 5
b)\(\lim_{x \to 0}\) \(\frac{1- \sqrt{x+1} }{x}\)=\(\lim_{x \to 0}\) \(\frac{1- \sqrt{x+1} }{x}( \frac{1+ \sqrt{x+1} }{1+ \sqrt{x+1} })\)=\(\lim_{x \to 0} \frac{-x}{x(1+ \sqrt{x+1} }\)=]=\(\lim_{x \to 0} \frac{-1}{(1+ \sqrt{x+1} }\) i teraz z x podstawić 0

[radagast]

c)
\(\lim_{x \to 0^-}\) \(\frac{x+1}{1+e^{ \frac{1}{x}} }=\frac{0+1}{1+e^{- \infty }}= \frac{0+1}{1+0} =1\)

[radagast]

d)
\(\lim_{x \to 0^+}\frac{x+1}{1+e^{ \frac{1}{x}} }=\lim_{x \to 0^+}\frac{0+1}{1+e^{+ \infty }}= \frac{1}{ \infty } =0\)