Download App

Limits — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSLimits3 Marks
Question
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}(\cos\text{x}+\text{a}\sin\text{bx})^\frac{1}{\text{x}}$
✓

Answer

$\lim\limits_{\text{x}\rightarrow0}(\cos\text{x}+\text{a}\sin\text{bx})^\frac{1}{\text{x}}$
$=\lim\limits_{\text{x}\rightarrow0}(1+(\cos\text{x}+\text{a}\sin\text{bx}))^\frac{1}{\text{x}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow0}\frac{(\cos\text{x}+\text{a}\sin\text{bx}-1)}{\text{x}}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow0}\frac{(\text{a}\sin\text{bx}-(1-\cos\text{x}))}{\text{x}}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow0}\frac{\big(\text{a}\sin\text{bx}-2\sin^2\big(\frac{\text{x}}{2}\big)\big)}{\text{x}}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow0}\frac{\text{ab}\sin\text{bx}}{\text{bx}}-\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\big(\frac{\text{x}}{2}\big)\sin\big(\frac{\text{x}}{2}\big)}{2\big(\frac{\text{x}}{2}\big)}}$
$=\text{e}^{\lim\limits_{\text{x}\rightarrow0}\frac{\text{ab}\sin\text{bx}}{\text{bx}}-\lim\limits_{\text{x}\rightarrow0}\frac{2\sin\big(\frac{\text{x}}{2}\big)\sin\big(\frac{\text{x}}{2}\big)}{\big(\frac{\text{x}}{2}\big)}}$
$=\text{e}^{\text{ab}-0}$
$=\text{e}^{\text{ab}}$

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 "3 Marks Question" from Limits→Limits — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Solve for x, the inequalities Exercises.
$\frac{|\text{x}-2|-1}{|\text{x}-2|-2}\leq0$ 
→
Find the equation of the circle passing through the points:
(0, 0), (-2, 1) and (-3, 2)
→
Prove that the points (0, -1, -7), (2, 1, -9) and (6, 5, -13) are collinear. Find the ratio in which the first point divides the join of the other two.
→
Find the mean deviation about the median for the data in: 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17.
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\tan8\text{x}}{\sin2\text{x}}$
→
Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(a, b): a, b $ \in $A, b is exactly divisible by a}.
  1. Write R in roster form
  2. Find the domain of R
  3. Find the range of R.
→
If the letters of the word ALGORITHM are arranged at random in a row what is the probability the letters GOR must remain together as a unit?
→
Show that the coefficient of the middle term in the expansion of $(1+ x )^{2 n }$ is equal to the sum of the coefficients of middle terms in the expansion of $(1+x)^{2 n -1}$.
→
Let f and g be two real functions defined by $\text{f(x)}=\sqrt{\text{x}+1}$ and $\text{g(x)}=\sqrt{9-\text{x}^2}$ Then describe the following functions:
f + g
→
Prove that:
$\cot^2\text{x}-\tan^2\text{x}=4\cot2\text{x}\ \text{cosec}\ 2\text{x}$
→