Limiti i funksionit Matura 2013. Forma e pacaktuar ∞-∞,∞/∞,0/0

Gjeni limitet:

a)$\lim_{x\rightarrow&space;-\infty&space;}\left&space;(&space;\sqrt{x^{2}+1}+x&space;\right&space;)$

b) $\lim_{x\rightarrow&space;0}\frac{\sqrt{x^{2}+1}-1}{\sqrt{x^{2}+16}-4}$

c) $\lim_{x\rightarrow&space;+\infty&space;}\frac{\sqrt{x^{2}+1}+\sqrt{x}}{\sqrt[4]{x^{3}+x}-x}$

zgjidhje

a) $\lim_{x\rightarrow&space;-\infty&space;}\left&space;(&space;\sqrt{x^{2}+1}+x&space;\right&space;)=\lim_{x\rightarrow&space;-\infty&space;}\frac{\left&space;(&space;\sqrt{x^{2}+1}+x&space;\right&space;)\left&space;(&space;\sqrt{x^{2}+1}-x&space;\right&space;)}{\left&space;(&space;\sqrt{x^{2}+1}-x&space;\right&space;)}=\lim_{x\rightarrow&space;-\infty&space;}\frac{x^{2}+1-x^{2}}{\sqrt{x^{2}+1}-x}=\lim_{x\rightarrow&space;-\infty&space;}\frac{1}{\sqrt{x^{2}+1}-x}=0$  sepse $\lim_{x\rightarrow&space;-\infty&space;}\left&space;(&space;\sqrt{x^{2}+1}-x&space;\right&space;)=\lim_{x\rightarrow&space;-\infty&space;}\left&space;(&space;\sqrt{x^{2}\left&space;(&space;1+\frac{1}{x^{2}}&space;\right&space;)}-x&space;\right&space;)=\lim_{x\rightarrow&space;-\infty&space;}\left&space;[&space;-x\left&space;(&space;\sqrt{1+\frac{1}{x^{2}}}+1&space;\right&space;)&space;\right&space;]&space;=+\infty$       $\sqrt{x^{2}}=-x$  per $x<0$

b)$\lim_{x\rightarrow&space;0}\frac{\sqrt{x^{2}+1}-1}{\sqrt{x^{2}+16}-4}=\lim_{x\rightarrow&space;0}\frac{\left&space;(&space;\sqrt{x^{2}+1}-1&space;\right&space;)\left&space;(&space;\sqrt{x^{2}+1}+1&space;\right&space;)\left&space;(\sqrt{x^{2}+16}+4&space;\right&space;)}{\left&space;(&space;\sqrt{x^{2}+16}-4&space;\right&space;)\left&space;(&space;\sqrt{x^{2}+16}+4&space;\right&space;)\left&space;(&space;\sqrt{x^{2}+1}+1&space;\right&space;)}=\lim_{x\rightarrow&space;0}\frac{x^{2}\left&space;(&space;\sqrt{x^{2}+16}+4&space;\right&space;)}{x^{2}\left&space;(&space;\sqrt{x^{2}+1}+1&space;\right&space;)}=\frac{8}{4}=4$

c) $\lim_{x\rightarrow&space;+\infty&space;}\frac{\sqrt{x^{2}+1}+\sqrt{x}}{\sqrt[4]{x^{3}+x}-x}=\lim_{x\rightarrow&space;+\infty&space;}\frac{x\left&space;(&space;\sqrt{1+\frac{1}{x^{2}}}+\sqrt{\frac{1}{x}}&space;\right&space;)}{x\left&space;(&space;\sqrt[4]{\frac{1}{x}+\frac{1}{x^{3}}}-1&space;\right&space;)}=-1$  Kryej veprimet qe nga faktorizimi brenda rrenjeve, pastaj thjeshto me x .

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