Question:
Evaluate the Given limit: \(\lim_{x\rightarrow 0}\) \(\frac{ax+xcos\,x}{b\,sin\,x}\)
Evaluate the Given limit: \(\lim_{x\rightarrow 0}\) \(\frac{ax+xcos\,x}{b\,sin\,x}\)
Updated On: Oct 23, 2023
Hide Solution
Verified By Collegedunia
Solution and Explanation
\(\lim_{x\rightarrow 0} \frac{ax+xcos\,x}{b\,sin\,x}\)
At x = 0, the value of the given function takes the form 0/0.
\(\lim_{x\rightarrow 0}\) \(\frac{cos^2x-1}{cos\,x-1}\) =\(\frac{1}{b}\)\(\lim_{x\rightarrow 0}\) \(\frac{x(a+cos\,x)}{sin\,x}\)
=\(\frac{1}{b}\)\(\times\)\(\lim_{x\rightarrow 0}\)(\(\frac{x}{sin\,x}\)) \(\times\)\(\lim_{x\rightarrow 0}\) (a + cosx)
=\(\frac{1}{b}\) \(\times\) (\(\lim_{x\rightarrow 0}\)\(\frac{1}{\frac{sin\,x}{x}}\)) \(\times\) \(\lim_{x\rightarrow 0}\) (a + cosx)
= \(\frac{1}{b}\) \(\times\) (a + cos0) [\(\lim_{x\rightarrow 0}\) \(\frac{sin\,x}{x}\) = 1]
= \(a+\frac{1}{b}\)
At x = 0, the value of the given function takes the form 0/0.
\(\lim_{x\rightarrow 0}\) \(\frac{cos^2x-1}{cos\,x-1}\) =\(\frac{1}{b}\)\(\lim_{x\rightarrow 0}\) \(\frac{x(a+cos\,x)}{sin\,x}\)
=\(\frac{1}{b}\)\(\times\)\(\lim_{x\rightarrow 0}\)(\(\frac{x}{sin\,x}\)) \(\times\)\(\lim_{x\rightarrow 0}\) (a + cosx)
=\(\frac{1}{b}\) \(\times\) (\(\lim_{x\rightarrow 0}\)\(\frac{1}{\frac{sin\,x}{x}}\)) \(\times\) \(\lim_{x\rightarrow 0}\) (a + cosx)
= \(\frac{1}{b}\) \(\times\) (a + cos0) [\(\lim_{x\rightarrow 0}\) \(\frac{sin\,x}{x}\) = 1]
= \(a+\frac{1}{b}\)
Was this answer helpful?
0
0
Top Questions on Limits
- If \[ \lim_{x \to \infty} \left( \frac{e}{1 - e} \left( \frac{1}{e} - \frac{x}{1 + x} \right) \right)^x = \alpha, \] then the value of \[ \frac{\log_e \alpha}{1 + \log_e \alpha} \] equals:
- Let \[ f(x) = \lim_{n \to \infty} \sum_{r=0}^{n} \left( \frac{\tan \left( \frac{x}{2^{r+1}} \right) + \tan^3 \left( \frac{x}{2^{r+1}} \right)}{1 - \tan^2 \left( \frac{x}{2^{r+1}} \right)} \right) \] Then, \( \lim_{x \to 0} \frac{e^x - e^{f(x)}}{x - f(x)} \) is equal to:
- If \[ \lim_{x \to \infty} \left( \frac{e}{1 - e} \left( \frac{1}{e} - \frac{x}{1 + x} \right) \right)^x = \alpha, \] then the value of \[ \frac{\log_e \alpha}{1 + \log_e \alpha} \] equals:
- Evaluate \( \int_{0}^{\pi/4} \frac{\cos^2 x}{\cos^2 x + 4 \sin^2 x} \, dx \):
- $\lim_{n \to \infty} \left(\frac{n}{n^2 + 1^2} + \frac{n}{n^2 + 2^2} + \dots + \frac{n}{n^2 + 3^2 + \dots + \frac{1}{5n}} \right) =$
View More Questions
Questions Asked in CBSE Class XI exam
- Consider the reactions:
- H3PO2(aq) + 4 AgNO3(aq) + 2 H2O(l) \(\rightarrow\) H3PO4(aq) + 4Ag(s) + 4HNO3(aq)
- H3PO2(aq) + 2CuSO4(aq) + 2 H2O(l) \(\rightarrow\) H3PO4(aq) + 2Cu(s) + H2SO4(aq)
- C6H5CHO(l) + 2[Ag (NH3)2]+(aq) + 3OH- (aq) \(\rightarrow\) C6H5COO- (aq) + 2Ag(s) + 4NH3(aq) + 2 H2O(l)
- C6H5CHO(l) + 2Cu2+(aq) + 5OH-(aq) \(\rightarrow\) No change observed.
What inference do you draw about the behaviour of Ag+ and Cu2+ from these reactions?
- CBSE Class XI
- Redox Reactions In Terms Of Electron Transfer Reactions
- Find \((x + 1)^6 - (x - 1)^6\). Hence or otherwise evaluate \((\sqrt2 + 1)^6 + (\sqrt2 - 1)^6.\)
- CBSE Class XI
- Binomial Theorem for Positive Integral Indices
- If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets:
(i) A = {a, b, c}
(ii) B = {d, e, f, g}
(iii) C = {a, c, e, g}
(iv) D = {f, g, h, a}- CBSE Class XI
- Complement of a Set
- Consider the following species: N3- , O2-, F-, Na+, Mg2+ and Al3+
- What is common in them?
- Arrange them in the order of increasing ionic radii.
- CBSE Class XI
- Modern Periodic Law And The Present Form Of The Periodic Table
- Find and write down structures of 10 interesting small molecular weight biomolecules. Find if there is any industry which manufactures the compounds by isolation. Find out who are the buyers.
- CBSE Class XI
- How To Analyse Chemical Composition?
View More Questions