Show that lim(x→∞)(√(x^2+ x + 1) - x) ≠ lim(√(x^2 + 1) - x)

Question

Show that \(\lim\limits_{\text x \to \infty}(\sqrt{\text x^2+\text x +1}-\text x) \) ≠ \(\lim\limits_{\text x \to \infty}(\sqrt{\text x^2+1}-\text x) \)

lim(x→∞)(√(x²+ x + 1) - x) ≠ lim(√(x² + 1) - x)

 To Prove:   \(\lim\limits_{\text x \to \infty}(\sqrt{\text x^2+\text x +1}-\text x) \) ≠ \(\lim\limits_{\text x \to \infty}(\sqrt{\text x^2+1}-\text x) \)

Answer 1

We have L.H.S  =  \(\lim\limits_{\text x \to \infty}(\sqrt{\text x^2+\text x +1}-\text x) \)

Rationalizing the numerator, we get,

Taking x as common from both numerator and denominator,

Now x→∞ and \(\cfrac1{\text x}\) = 0 then

Therefore, R.H.S = 0

So, L.H.S ≠ R.H.S

Hence,   \(\lim\limits_{\text x \to \infty}(\sqrt{\text x^2+\text x +1}-\text x) \) ≠ \(\lim\limits_{\text x \to \infty}(\sqrt{\text x^2+1}-\text x) \)