Evaluate the following limit: _ x x x^2+1 - x^2-1 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow\infty}\text{x}\Big\{\sqrt{\text{x}^2+1}-\sqrt{\text{x}^2-1}\Big\}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow\infty}\text{x}\Big\{\sqrt{\text{x}^2+1}-\sqrt{\text{x}^2-1}\Big\}$
$=\lim\limits_{\text{x}\rightarrow\infty}\text{x}\Big[\sqrt{\text{x}^2+1}-\sqrt{\text{x}^2-1}\Big]\times\frac{\big(\sqrt{\text{x}^2+1}+\sqrt{\text{x}^2-1}\big)}{\big(\sqrt{\text{x}^2+1}+\sqrt{\text{x}^2+1}\big)}$
$=\lim\limits_{\text{x}\rightarrow\infty}\text{x}\frac{\text{x}\big(\text{x}^2+1-\text{x}^2+1\big)}{\sqrt{\text{x}^2+1}+\sqrt{\text{x}^2-1}}$
$=\lim\limits_{\text{x}\rightarrow\infty}\frac{\text{x}(2)}{\text{x}\Big(\sqrt{1+\frac{1}{\text{x}^2}}+\sqrt{1-\frac{1}{\text{x}^2}}\Big)}$
$=\lim\limits_{\text{x}\rightarrow{\infty}}\frac{2}{\sqrt{1+\frac{1}{\text{x}^2}}+\sqrt{1-\frac{1}{\text{x}^2}}}$
$=\frac{2}{2}=1$
$=1$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Limits→Limits — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Prove the following by the principle of mathematical induction:$\text{a}+(\text{a}+\text{d})+(\text{a}+2\text{d})...+(\text{a}+(\text{n}-1)\text{d})=\frac{\text{n}}{2}[2\text{a}+(\text{n}-1)\text{d}]$

If the sides a, b, c of a $\triangle\text{ABC}$ are in H.P., prove that $\sin^2\frac{\text{A}}{2},\sin^2\frac{\text{B}}{2},\sin^2\frac{\text{C}}{2}$ are in H.P.

Find the equation of the straight lines passing through the following pair of points:
$(\text{a}\cos\alpha, \ \text{a} \ \sin\alpha)$ and $(\text{a}\cos\beta, \ \text{a} \ \sin\beta)$

Differentiate the functions with respect to 'x'.
$\text{x}^{\frac{2}{3}}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\sin\sqrt{\text{x}}-\sin\sqrt{\text{a}}}{\text{x}-\text{a}}$

Determine the domain and range of the relation R defined by:
$\text{R}=\{(\text{x, x,}+5):\text{x}\in\{0,1,2,3,4,5\}\}$

If $\text{x}^{\text{a}}=\text{x}^{\frac{\text{b}}{2}}\text{z}^{\frac{\text{b}}{2}}=\text{z}^{\text{c}},$ then prove that $\frac{1}{\text{a}},\frac{1}{\text{b}},\frac{1}{\text{c}}$ are in A.P.

In a $\triangle\text{ABC},$ prove that
$\sin^3\text{A}\cos(\text{B}-\text{C})+\sin^2\text{B}\cos(\text{C}-\text{A})+\sin^3\text{C}\cos(\text{A}-\text{B})\\=3\sin\text{A}\sin\text{B}\sin\text{C}$

Prove that:
$\frac{\sin\text{A}\sin2\text{A}+\sin3\text{A}\sin6\text{A}}{\sin\text{A}\cos2\text{A}+\sin3\text{A}\cos6\text{A}}=\tan5\text{A}$

Sketch the graphs of the following trigonometric functions:
$\text{h(x)}=\cos^22\text{x}$