Download App

Limits — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsLimits5 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{3}}}\frac{\sqrt{3}-\tan\text{x}}{\pi-3\text{x}}$
✓

Answer

$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{3}}}\frac{\sqrt{3}-\tan\text{x}}{\pi-3\text{x}}$
If $\text{x}\rightarrow\frac{\pi}{3},\frac{\pi}{3}-\text{x}\rightarrow0,\pi-3\text{x}\rightarrow0$
Let $\frac\pi3-\text{x}=\text{y}$ they y → 0
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\sqrt{3}-\tan\big(\frac{\pi}{3}-\text{y}\big)}{3\big(\frac{\pi}{3}-\text{x}\big)}$
$=\lim\limits_{\text{y}\rightarrow{0}}\begin{pmatrix}\frac{\bigg(\sqrt{3}-\frac{\tan\frac\pi3-\tan\text{y}}{1+\tan\frac\pi3.\tan\text{y}}\bigg)}{3\text{y}}\end{pmatrix}$
$=\lim\limits_{\text{y}\rightarrow{0}}\begin{pmatrix}\frac{\Big(\sqrt{3}-\frac{\sqrt{3}-\tan\text{y}}{1+\sqrt{3}\tan\text{y}}\Big)}{3\text{y}}\end{pmatrix}$
$=\lim\limits_{\text{y}\rightarrow{0}}\frac{\big(\sqrt{3}-\tan\text{y}-\sqrt{3}+\tan\text{y}\big)}{3\big(1+\sqrt{3}\tan\text{y}\big)\text{y}}$
$=\frac{4}{3}\times\lim\limits_{\text{y}\rightarrow0}\frac{\tan\text{y}}{\text{y}}\times\frac{1}{\lim\limits_{\text{y}\rightarrow0}\Big(1+\sqrt{3}\frac{\tan\text{y}}{\text{y}}\times\text{y}\Big)}$
$=\frac{4\times1}{3}\times\frac{1}{1+0}$
$=\frac{4}{3}$

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 "5 Marks Questions" from Limits→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 $ \tan\text{A}+\tan\text{B}=\text{a}$ and $\cot\text{A}+\cot\text{B}=\text{b},$ prove that $\cot\text{(A}+\text{B)}=\frac{1}{\text{a}}-\frac{1}{\text{b}}.$
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow1}(1-\text{x})\tan\Big(\frac{\pi\text{x}}{2}\Big)$
→
Evaluate the following limit:
$\lim\limits_{\text{n}\rightarrow\infty}\frac{1^3+2^3+\ \dots+\text{n}^3}{(\text{n}-1)^4}$
→
Sketch the graphs of the following trigonometric functions:
$\text{h(x)}=\cos^22\text{x}$
→
Show that the solution set of the following system of linear inequalities is an unbounded region: 2x + y > 8, x + 2y > 10, x > 0, y > 0.
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow-\infty}\big(\sqrt{4\text{x}^2-7\text{x}}+2\text{x}\big)$
→
Solve the following equations:
$2\sin^{2}\text{x}=3\cos\text{x},0\leq\text{x}\leq2\pi$
→
Two ships leave a port at the same time. One goes 24km/ hr in the direction N 38° E and other travels 32km/ hr in the direction S 52° E. Find the distance between the ships at the end of 3hrs.
→
Evaluate the following:
$(\sqrt3+\sqrt2)^6-(\sqrt3-\sqrt2)^6$
→
Find the number of words formed by permuting all the letters of the following words:
EXERCISES.
→