Evaluate the following limit: _ x 2 5+2x - (2+1 ) x^2-2 - Vidyadip

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\sqrt{2}}}\frac{\sqrt{5+2\text{x}}-\big(\sqrt{2}+1\big)}{\text{x}^2-2}$

✓

Answer

$\lim\limits_{\text{x}\rightarrow{\sqrt{2}}}\frac{\sqrt{5+2\text{x}}-\big(\sqrt{2}+1\big)}{\text{x}^2-2}$$=\lim\limits_{\text{x}\rightarrow{\sqrt{2}}}\frac{\sqrt{3+2\text{x}}-\Big(\sqrt{\big(\sqrt{2}+1\big)^2}\Big)}{\text{x}^2-2}$
$=\lim\limits_{\text{x}\rightarrow{\sqrt{2}}}\frac{\sqrt{3+2\text{x}}-\big(\sqrt{3+2\sqrt{2}}\big)}{\text{x}^2-2}$
$=\lim\limits_{\text{x}\rightarrow{\sqrt{2}}}\frac{\sqrt{3+2\text{x}}-\big(\sqrt{3+2\sqrt{2}}\big)}{\text{x}^2-2}\times\frac{\sqrt{3+2\text{x}}+\big(\sqrt{3+2\sqrt{2}}\big)}{\sqrt{3+2\text{x}}+\big(\sqrt{3+2\sqrt{2}}\big)}$
$=\lim\limits_{\text{x}\rightarrow{\sqrt{2}}}\frac{{3+2\text{x}}-{3-2\sqrt{2}}}{\big(\text{x}^2-2\big)\Big(\sqrt{3+2\text{x}}+\big(\sqrt{3+2\sqrt{2}}\big)\Big)}$
$=\lim\limits_{\text{x}\rightarrow{\sqrt{2}}}\frac{2\big(\text{x}-\sqrt{2}\big)}{\big(\text{x}^2-2\big)\Big(\sqrt{3+2\text{x}}+\big(\sqrt{3+2\sqrt{2}}\big)\Big)}$
$=\lim\limits_{\text{x}\rightarrow{\sqrt{2}}}\frac{2}{\big(\text{x}^2-\sqrt{2}\big)\Big(\sqrt{3+2\text{x}}+\big(\sqrt{3+2\sqrt{2}}\big)\Big)}$
$=\frac{2}{\big(\sqrt{2}+\sqrt{2}\big)\Big(\sqrt{3+2\sqrt{2}}+\big(\sqrt{3+2\sqrt{2}}\big)\Big)}$
$=\frac{2}{\big(2\sqrt{2}\big)\Big(2\sqrt{3+2\sqrt{2}}\Big)}$
$=\frac{1}{\big(2\sqrt{2}\big)\Big(\sqrt{3+2\sqrt{2}}\Big)}$
$=\frac{1}{\big(2\sqrt{2}\big)\Big(\sqrt{2}+1\Big)}$
$=\frac{\big(\sqrt{2}-1\big)}{\big(2\sqrt{2}\big)}$

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 4 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

Prove that $\frac{(2\text{n})!}{2^{2\text{n}}(\text{n}!)^2}\leq\frac{1}{\sqrt{3\text{n}+1}}$ for all $\text{n}\in\text{N}.$

In a $\triangle\text{ABC},$ if $\angle\text{B}=60^{\circ},$ prove that (a + b + c) = 3ca

Solve the following system of equations in R. $2\text{x}+5\leq0,\text{x}-3\leq0$

Find the image of the point (2, 1) with respect to the line mirror x + y - 5 = 0.

Let r and n be positive integers such that 1 < r < n. Then prove the following: $\frac{{{^\text{n}}\text{C}_{\text{r}}}}{{^\text{n-1}}\text{C}_{\text{r}-1}}=\frac{\text{n}}{\text{r}}$

How many triangles can be obtained by joining 12 points, five of which are collinear?

Prove that the area of the parallelogram formed by the lines 3x - 4y + a = 0, 3x - 4y + 3a= 0, 4x - 3y - a = 0 and 4x - 3y - 2a = 0 is $\frac{2\text{a}^2}{7}$ sq.units.

An analysis of the weekly wages paid to workers in two firms A and B, belonging to the same industry gives the following results:

	Firm A	Firm B
No. of wage earners	586	648
Average weekly wages	52.5	47.5
Variance of the	100	121
Distribution of wages

Which firm A or B pays out larger amount as weekly wages?
Which firm A or B has greater variability in individual wages?

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow1}\frac{\sqrt{3+\text{x}}-\sqrt{5-\text{x}}}{\text{x}^2-1}$

The mean and standard deviation of six observation are 8 and 4 respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.