Evaluate the following limit: _ x 0 1+x - 1-x 2x - Vidyadip

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{1+\text{x}}-\sqrt{1-\text{x}}}{2\text{x}}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{1+\text{x}}-\sqrt{1-\text{x}}}{2\text{x}}$$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{1+\text{x}}-\sqrt{1-\text{x}}\big)}{2\text{x}}\times\frac{\big(\sqrt{1+\text{x}}+\sqrt{1-\text{x}}\big)}{\big(\sqrt{1+\text{x}}+\sqrt{1-\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{(1+\text{x})-{(1-\text{x}})}{2\text{x}\big(\sqrt{1+\text{x}}+\sqrt{1-\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{2\text{x}}{2\text{x}\big(\sqrt{1+\text{x}}+\sqrt{1-\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{1}{\big(\sqrt{1+\text{x}}+\sqrt{1-\text{x}}\big)}$
$=\frac{1}{\sqrt{1}+\sqrt{1}}$
$=\frac{1}{2}$

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 rational number having the following decimal expansions:$0.\overline{3}$

Find the value of x for which the points (x, -1), (2, 1) and (4, 5) are collinear.

From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them, will join or none of them will join. In how many ways can the excursion party be chosen?

Differentiate the following function with respect to x:$\text{e}^\text{x}\log\sqrt{\text{x}}\tan\text{x}$

How many different words can be formed with the letters of word 'SUNDAY'? How many of the words begin with N? How many begin with N and end in Y?

Show the following quadratic equation: $2 x^2-(3+7 i) x+(9 i-3)=0$

Solve the inequality and show the graph for the solution on number line: 3 (1 – x) < 2 (x + 4)

Find the equation of the straight line on which the length of the perpendicular from the origin makes an angle of 30° with x-axis and which forms a triangle of area $\frac{50}{\sqrt3}$ with the axes.

$\text{a}^2(\cos^2\text{B}-\cos^2\text{C})+\text{b}^2(\cos^2\text{C}-\cos^2\text{A})\\+\text{c}^2(\cos^2\text{A}-\cos^2\text{B})=0$

Find the $20^{th}$ term of the series $2 \times 4 + 4 \times 6 + 6 \times 8 + \ldots \ldots \ldots +$ n terms.