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

Question

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

✓

Answer

$\lim\limits_{\text{x}\rightarrow{\infty}}{\sqrt{\text{x}+1}-\sqrt{\text{x}}}{}$ $=\lim\limits_{\text{x}\rightarrow{\infty}}\frac{\big(\sqrt{\text{x}+1}-\sqrt{\text{x}}\big)\big(\sqrt{\text{x}+1}+\sqrt{\text{x}}\big)}{\big(\sqrt{\text{x}+1}+\sqrt{\text{x}}\big)}$ $=\lim\limits_{\text{x}\rightarrow{\infty}}\frac{\big(\text{x}+1-\text{x}\big)}{\sqrt{\text{x}+1}+\sqrt{\text{x}}}$ $=\lim\limits_{\text{x}\rightarrow{\infty}}\bigg(\frac{1}{\sqrt{\text{x}+1}+\sqrt{\text{x}}}\bigg)$ $=\frac{1}{\infty}$ $=0$

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 "(Each question 3 marks)" from Limits→Limits — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the equation of the ellipse whose vertices are ($\pm$ 13, 0) and foci are ($\pm$ 5, 0).

The hypotenuse of a right-angled triangle has its ends at the points (1, 3) and (- 4, 1). Find the equation of the legs (perpendicular sides) of the triangle.

Let f : $R^+ → R,$ where $R^+$ is the set of all positive real numbers, such that $\text{f(x)}=\log_\text{e}\text{x}.$ Determine:

The image set of the domain of f
{x : f(x) = -2}
Whether f(xy) = f(x) + f(y) holds.

Find the equation of the circle which circumscribes the triangle formed by the lines $x + y + 3 = 0, x - y + 1 = 0$ and $x = 3$

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{2\text{x}}{\sqrt{\text{a}+\text{x}}-\sqrt{\text{a}-\text{x}}}$

Prove that $:\frac{\text{cosec}(90^{\circ}+\text{x})+\cot(450^\circ+\text{x})}{\text{cosec}(90^\circ-\text{x})+\tan(180^\circ-\text{x})} +\frac{\tan\text{x}(180^\circ+\text{x})+\sec(180^\circ-\text{x})}{\tan(360^\circ+\text{x})-\sec(-\text{x})}=2$

If $\cos\text{A}=-\frac{24}{25},$ $\cos\text{B}=-\frac{12}{13},$ where A and B both lie in second quadrant,find the value of $\sin\text{(A+B)}$.

$\sin\text{(A+B)}$
$\cos\text{(A+B)}$

Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23.

Find the multiplicative inverse of the following complex numbers: $4-3\text{i}$

If ${^\text{2n}}\text{C}_{\text{3}}:{^\text{n}}\text{C}_{\text{2}}=44:3,$ find n.