Bonjour
Yacine931
[tex]1)\ \lim\limits_{t\to+\infty}\dfrac{1}{t}=0\\\\\Longrightarrow\lim\limits_{t\to+\infty}\sqrt{4+\dfrac{1}{t}}=\sqrt{4+0}=\sqrt{4}=2\\\\\Longrightarrow\boxed{\lim\limits_{t\to+\infty}\sqrt{4+\dfrac{1}{t}}=2}[/tex]
[tex]2)\ \lim\limits_{t\to0^+}\dfrac{1}{t}=+\infty\\\\\Longrightarrow\lim\limits_{t\to0^+}(4+\dfrac{1}{t})=+\infty\\\\\Longrightarrow\boxed{\lim\limits_{t\to0^+}\sqrt{4+\dfrac{1}{t}}=+\infty}[/tex]
[tex]3)\ \lim\limits_{x\to0}\dfrac{\cos x-1}{x}=\lim\limits_{x\to0}\dfrac{(\cos x-1)(\cos x+1)}{x(\cos x+1)}\\\\\\=\lim\limits_{x\to0}\dfrac{\cos^2 x-1}{x(\cos x+1)}\\\\\\=\lim\limits_{x\to0}\dfrac{-\sin^2 x}{x(\cos x+1)}\\\\\\=\lim\limits_{x\to0}[\dfrac{\sin x}{x}\times\sin x\times(\dfrac{-1}{\cos x+1})]\\\\\\=\lim\limits_{x\to0}\dfrac{\sin x}{x}\times\lim\limits_{x\to0}\sin x\times\lim\limits_{x\to0}(\dfrac{-1}{\cos x+1})\\\\\\=1\times\sin 0\times(\dfrac{-1}{\cos 0+1})[/tex]
[tex]\\\\\\=1\times0\times(\dfrac{-1}{1+1})\\\\=0\\\\\Longrightarrow\boxed{\lim\limits_{x\to0}\dfrac{\cos x-1}{x}=0}[/tex]
[tex]4)\ \lim\limits_{x\to+\infty}(x+1-\sqrt{x^2+2x+2})\\\\\\=\lim\limits_{x\to+\infty}\dfrac{(x+1-\sqrt{x^2+2x+2})(x+1+\sqrt{x^2+2x+2})}{x+1+\sqrt{x^2+2x+2}}\\\\\\=\lim\limits_{x\to+\infty}\dfrac{(x+1)^2-(\sqrt{x^2+2x+2})^2}{x+1+\sqrt{x^2+2x+2}}\\\\\\=\lim\limits_{x\to+\infty}\dfrac{(x^2+2x+1)-(x^2+2x+2)}{x+1+\sqrt{x^2+2x+2}}\\\\\\=\lim\limits_{x\to+\infty}\dfrac{x^2+2x+1-x^2-2x-2}{x+1+\sqrt{x^2+2x+2}}\\\\\\=\lim\limits_{x\to+\infty}\dfrac{-1}{x+1+\sqrt{x^2+2x+2}}=[-\dfrac{1}{+\infty}]\\\\=0[/tex]
[tex]\Longrightarrow\boxed{\lim\limits_{x\to+\infty}(x+1-\sqrt{x^2+2x+2})=0}[/tex]
