Question

Evaluate: lim x -> ∞ (sqrt(x ^ 2 + 1))/(x + 1)​

Answer 1

`\lim_(x\rightarrow +\infty ) \frac{\sqrt{x^(2)+1} }{x+1}=1`

Explanation:

`\lim_(x\rightarrow +\infty ) \frac{\sqrt{x^(2)+1} }{x+1}`

`=\lim_(x\rightarrow +\infty ) \frac{\sqrt{x^(2)\left( 1+(1)/(x^2) \right) } }{x+1}`

`=\lim_(x\rightarrow +\infty ) \frac{\sqrt{x^(2)} \sqrt{\left( 1+(1)/(x^2) \right) } }{x+1}`

`=\lim_(x\rightarrow +\infty ) \frac{|x|\sqrt{\left( 1+(1)/(x^2) \right) } }{x+1}`

`=\lim_(x\rightarrow +\infty ) \frac{x\sqrt{\left( 1+(1)/(x^2) \right) } }{x(1+(1)/(x)) }`

`=\lim_(x\rightarrow +\infty ) \frac{\sqrt{\left( 1+(1)/(x^2) \right) } }{(1+(1)/(x)) }`

`=(√(1) )/(1)`

`=1`

Remark :

`\lim_(x\rightarrow +\infty ) (1)/(x) =0=\lim_(x\rightarrow +\infty ) (1)/(x^2)`