granice

[malineczka8888]

oblicz:
a) \(\lim_{x \to 0}\) \(\frac{sin4x}{sin7x}\)
b) \(\lim_{x \to \pi}\) \(\frac{sinx}{x- \pi }\)
c) \(\lim_{x \to 1}\) \(\frac{sin(1-x)}{ \sqrt{x}-1 }\)
d) \(\lim_{x\to \infty}\) \((1+ \frac{1}{x})^{x+1}\)
e) \(\lim_{x\to \infty}\) \((\frac{x+1}{x-2})^{2x}\)

[Lbubsazob]

\(\lim_{x\to 0} \frac{\sin 4x}{\sin 7x}= \lim_{x\to 0} \sin 4x \cdot \frac{4x}{4x} \cdot \frac{1}{\sin 7x} \cdot \frac{7x}{7x}= \lim_{x\to 0} \frac{4x}{7x}=\frac{4}{7}\)

[radagast]

b)
\(\lim_{x \to \pi}\) \(\frac{sinx}{x- \pi }= \lim_{x \to \pi}\) \(\frac{sin( \pi -x)}{x- \pi }= -\lim_{x \to \pi}\frac{sin( x- \pi )}{x- \pi }=-1\)

[radagast]

c)
\(\lim_{x \to 1}\) \(\frac{sin(1-x)}{ \sqrt{x}-1 }= \lim_{x \to 1}\) \(\frac{(\sqrt{x}+1 )sin(1-x)}{x-1 }=-\lim_{x \to 1}\) \(\frac{(\sqrt{x}+1 )sin(1-x)}{1-x }=-2\)

[radagast]

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

[radagast]

e)
\(\lim_{x\to \infty}\) \((\frac{x+1}{x-2})^{2x}=\lim_{x\to \infty}\) \((\frac{x-2+3}{x-2})^{2x}=\lim_{x\to \infty}\) \((1+\frac{3}{x-2})^{2(x-2)+4}=(e^3)^2=e^6\)