Evaluate _ z 1 z^ 1/3 - 1 z^ 1/6 - 1 - Vidyadip

Question

Evaluate $\mathop {\lim }\limits_{z \to 1} \frac{{{z^{1/3}} - 1}}{{{z^{1/6}} - 1}}$

✓

Answer

Here $\mathop {\lim }\limits_{z \to 1} \frac{{{z^{1/3}} - 1}}{{{z^{1/6}} - 1}}\left[ {\frac{0}{0}{\text{form}}} \right]$
$= \mathop {\lim }\limits_{z \to 1} \frac{{{{({z^{1/6}} - 1)}^2}}}{{{z^{1/6}} - 1}}$
$ = \mathop {\lim }\limits_{z \to 1} \frac{{({z^{1/6}} + 1)({z^{1/6}} - 1)}}{{({z^{1/6}} - 1)}} = \mathop {\lim }\limits_{z \to 1}$ = z^1/6 + 1
= (1)^1/6 + 1 = 1 + 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 "2 Marks Questions" from Limits and Derivatives→Limits and Derivatives — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the derivative of cos x from first principle.

Find the equation of the hyperbola with:
$\text{Vertices }(0,\pm7),\text{ e}=\frac{4}{3}$

Differentiate the following function with respect to (x):

$\text{x}^4-2\sin\text{x}+\cos\text{x}$

A coin is tossed three times, consider the events"
A: ‘No head appears’,
B: ‘Exactly one head appears’ and
C: ‘Atleast two heads appear’.
Do they form a set of mutually exclusive and exhaustive events?

Find the sun of all odd numbers between 100 and 200.

Find the equations of the straight lines which cut off an intercept 5 from the y-axis and are equally inclined to the axes.

Find the coefficient of:

x¹⁰ in the expansion of $\Big(2\text{x}^2-\frac{1}{\text{x}}\Big)^{20}.$

Prove that:
$\sec\Big(\frac{3\pi}{2}-\text{x}\Big)\sec\Big(\text{x}-\frac{5\pi}{2}\Big)+\tan\Big(\frac{5\pi}{2}+\text{x}\Big)\tan\Big(\text{x}-\frac{3\pi}{2}\Big)=-1$

Find the equation of the line perpendicular to the line x - 7y + 5 = 0 and having x intercept 3.

Find the domain and the range of the real function: $f(x)=\frac{1}{\sqrt{x^2-1}}$