Exercices d'entraînement sur les limites

\( \displaystyle \lim_{x \to 0 } \sqrt{1 + \dfrac1{x^2} } = + \infty \)

\(\displaystyle \lim_{x\rightarrow 0}\dfrac{1}{x^{2}}=+\infty\) , donc \(\displaystyle \lim_{x\rightarrow 0}\left(1+\dfrac{1}{x^{2}}\right)=+\infty\) et \(\displaystyle \lim_{X\rightarrow+\infty} \sqrt X=+\infty \), donc \( \displaystyle \lim_{x \to 0 } \sqrt{1 + \frac1{x^2} } = + \infty \).

\(\displaystyle \lim_{x \to 0}\sqrt{3+x^2}=\sqrt 3 \)

\(\displaystyle \lim_{x\to0}x^2=0\), donc \(\displaystyle \lim_{x\rightarrow{0}}\left(3+x^2\right)=3 \) et \(\displaystyle \lim_{X\rightarrow3} \sqrt X=\sqrt 3 \), donc \(\displaystyle \lim_{x \to 0 }\sqrt{3+x}=\sqrt 3 \).