Practice Questions for CSIR NET Real Analysis : Limit and Properties of Functions II

11. What are the values of \(a\) and \(b\) respectively, if \(\lim_{x\rightarrow 0} \frac{\sin ax - \ln(e^x\cos x)}{x\sin bx} = \frac{1}{2}\)?

(a.) 1,1
(b.) 1,\(\frac{1}{2}\)
(c.) -1,1
(d.) -1,\(\frac{1}{2}\)

12. Assertion (A): \(e^{x}\) cannot be expressed as the sum of even and odd functions

Reason (R): \(e^{x}\) is neither an even nor an odd function

(a.) Both A and R are individually true and R is the correct explanation of A
(b.) Both A and R are individually true but R is not the correct explanation of A
(c.) A is true but R is false
(d.) A is false but R is true

13. Given the following limits:

1. \(\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1\)

2. \(\lim_{x\rightarrow \infty} \left(1 + \frac{1}{x}\right) = 1\)

3. \(\lim_{x\rightarrow 0} \frac{\log(x+1)}{x} = 1\)

Which of these are correct?

(a.) 1, 2, and 3
(b.) 1 and 2
(c.) 1 and 3
(d.) 2 and 3

14. If \(f(x) = \frac{x-5}{x+5}\) for \(x \neq -5\), then the domain of \(f^{-1}(x)\) is

(a.) \(\mathbb{R}\)
(b.) \(\mathbb{R} - \{1\}\)
(c.) \((- \infty, 1)\)
(d.) \((1, \infty)\)

15. \(\lim_{x\rightarrow\infty} (4^x+5^x)^{\frac{1}{x}}\) equals

(a.) 4
(b.) 5
(c.) \(e\)
(d.) \(5e\)

16. \(\lim_{x\rightarrow 0} \frac{e^x+e^{-x}-2}{x^2}\) is equal to

(a.) 1
(b.) -1
(c.) \(\frac{1}{2}\)
(d.) \(-\frac{1}{2}\)

17. If \(\alpha\) and \(\beta\) are the roots of the equation \(ax^2+bx+c=0\), then \(\lim_{x\rightarrow \alpha} \frac{1-\cos(ax^2+bx+c)}{(x-\alpha)^2}\) is equal to

(a.) 0
(b.) \(\frac{a}{2}(\alpha-\beta)^2\)
(c.) \(-\frac{a}{2}(\alpha-\beta)^2\)
(d.) \(\frac{a^2}{2}(\alpha-\beta)^2\)

18. If \(\lim_{x\rightarrow 0} \frac{x(1-\cos x)-ax^2\sin x}{x^5}\) exists and is finite, then the value of \(a\) must be

(a.) 1
(b.) \(\frac{1}{2}\)
(c.) \(\frac{1}{3}\)
(d.) \(\frac{1}{4}\)

19. Which of the following statement is true?

(a.) \(\lim_{x\rightarrow\infty} \frac{\log x}{x^{1/2}} = 0\) and \(\lim_{x\rightarrow\infty} \frac{\log x}{x} = \infty\)
(b.) \(\lim_{x\rightarrow\infty} \frac{\log x}{x^{1/2}} = \infty\) and \(\lim_{x\rightarrow\infty} \frac{\log x}{x} = 0\)
(c.) \(\lim_{x\rightarrow\infty} \frac{\log x}{x^{1/2}} = 0\) and \(\lim_{x\rightarrow\infty} \frac{\log x}{x} = 0\)
(d.) \(\lim_{x\rightarrow\infty} \frac{\log x}{x^{1/2}} = 0\) but \(\lim_{x\rightarrow\infty} \frac{\log x}{x}\) does not exist

20. Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) define \(g: \mathbb{R} \rightarrow \mathbb{R}\) by \(g(x) = f(x)(f(x)+f(-x))\). Then

(a.) \(g\) is even for all \(f\)
(b.) \(g\) is odd for all \(f\)
(c.) \(g\) is even if \(f\) is even
(d.) \(g\) is even if \(f\) is odd