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

1. If \(f(x)=\frac{1}{1-x}\), \(g(x)=f[f(x)]\) and \(h(x)=f[g(x)]\), then what is \(f(x)g(x)h(x)\) equal to?

(a.) -1
(b.) 0
(c.) 1
(d.) 2

2. What is the value of \(\lim_{n\rightarrow\infty} \frac{2^{n+1}+3^{n+1}}{2^n+3^n}\)?

(a.) \(1/3\)
(b.) 1
(c.) 3
(d.) \(\infty\)

3. What is the value of \(\lim_{x\rightarrow 0} \frac{e^x-x-1}{x^2}\)?

(a.) 0
(b.) \(1/2\)
(c.) 2
(d.) \(e\)

4. If \(D\) is the set of all real \(x\) such that \(f(x)=1-e^{(1/x)-1}\) is positive, then what is \(D\) equal to?

(a.) \((-\infty, 0) \cup (1, \infty)\)
(b.) \((-\infty, 0) \cup [1, \infty)\)
(c.) \((1, \infty)\)
(d.) \((-\infty, 1)\)

5. If \(\lim_{x\rightarrow 0} \frac{x+3\sin x - x^3 - k\sinh x}{1-\cos x+x^2-3x^3}\) exists, then what is the value of \(k\)?

(a.) -1
(b.) 2
(c.) 3
(d.) 4

6. What is the value of \(\lim_{x\rightarrow y} \frac{x^y - y^x}{x^x - y^y}\)?

(a.) \(\frac{1+\ln y}{1-\ln y}\)
(b.) \(\frac{1-\ln y}{1+\ln y}\)
(c.) \(\frac{-1+\ln y}{1+\ln y}\)
(d.) \(\frac{-1-\ln y}{1-\ln y}\)

7. Assertion (A): \(\lim_{x\rightarrow\infty} \frac{\sin x}{x}=0\)

Reason (R): \(\lim_{x\rightarrow\infty} \frac{\sin x}{x} = \lim_{x\rightarrow\infty} \sin x \lim_{x\rightarrow\infty} \frac{1}{x}\)

Both A and R are individually true and R is the correct explanation of A
Both A and R are individually true but R is not the correct explanation of A
A is true but R is false
A is false but R is true

8. What is the value of \(\lim_{x\rightarrow\infty} \{\sin(1/x) + \cos(1/x)\}\)?

(a.) 0
(b.) \(\frac{1}{e}\)
(c.) \(e\)
(d.) 1

9. Which one of the following functions is not well defined?

(a.) \(\sqrt{1+\sin x}\)
(b.) \(\sqrt{5\sec^2 x - 4}\)
(c.) \(\sqrt{3-\cos^2 x}\)
(d.) \(\sqrt{x^4+x^2+\frac{1}{100}}\)

10. Which one of the following graphs is the correct graph of the function \(y(x) = x\ln(x)\)?

(a.)

(b.)

(c.)

(d.)