_ y 0 1 + 1 + y^4 - 2 y^4 = - Vidyadip

MCQ

$\mathop {\lim }\limits_{y \to 0} \frac{{\sqrt {1 + \sqrt {1 + {y^4}} } - \sqrt 2 }}{{{y^4}}} = $

✓
exists and equals $\frac{1}{{4\sqrt 2 }}$
B
exists and equals $\frac{1}{{2\sqrt 2 \left( {\sqrt 2 + 1} \right)}}$
C
exists and equals $\frac{1}{{2\sqrt 2 }}$
D
does not exist

✓

Answer

Correct option: A.

exists and equals $\frac{1}{{4\sqrt 2 }}$

a
${\left( {1 + x} \right)^n} \cong 1 + nx$ (when $x \to 0$)

So, $\sqrt {1 + {y^4}} = 1 + \frac{{{y^4}}}{2}$

$\mathop {\lim }\limits_{y \to 0} \frac{{\sqrt {2 + \frac{{{y^4}}}{2}} - \sqrt 2 }}{{{y^4}}}$

$ = \frac{{\sqrt 2 \left( {1 + \frac{{{y^4}}}{8} - 1} \right)}}{{{y^4}}} = \frac{{\sqrt 2 }}{8} = \frac{1}{{4\sqrt 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 "M.C.Q (1 Marks)" from STD 11 - 12. limits→STD 11 - 12. limits — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

If $\frac{1+7\text{i}}{(2-\text{i})^2},$ then:

The number of values of x in $[0,\ 2\pi]$ that satisfy the equation $\sin^2\text{x}-\cos\text{x}=\frac{1}{4}$

Set $A$ has $3$ elements and set $B$ has $6$ elements. What can be the minimum number of elements in $\ce{A ∪ B}\ ?$

The number of points of intersection of $| z -(4+3 i )|=2$ and $| z |+| z -4|=6, z \in C$ is

$\sin \frac{\pi}{12}=?$

The sum of two complex numbers $a + ib$ and $c + id$ is purely imaginary if

$\lim _{x \rightarrow 0} \frac{e^{2 |\text { sin } x | \mid}-2|\sin x|-1}{x^2}$

If three letters can be posted to any one of the $5$ different addresses, then the probability that the three letters are posted to exactly two addresses is:

If $\cos(\text{A}-\text{B})=\frac35$ and $\tan\text{A}\tan\text{B}=2,$ then

The ordinates of the points $P$ and $Q$ on the parabola with focus $(3,0)$ and directrix $x =-3$ are in the ratio $3: 1$. If $R (\alpha, \beta)$ is the point of intersection of the tangents to the parabola at $P$ and $Q$, then $\frac{\beta^2}{\alpha}$ is equal to $.............$.