Practice Questions for NET JRF Real Analysis Assignment: Limit and Properties of Functions III

21. If \(\lim_{x\rightarrow 0} \frac{\sin 2x + a\sin x}{x^3} = b\), where \(\lim_{x\rightarrow 0} \frac{\sin 2x + a\sin x}{x^3}\) is finite, then the values of \(a\) and \(b\) respectively will be

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

22. \(\lim_{x\rightarrow\infty} x \sin\left(\frac{1}{x}\right)\) equals

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

23. Consider the following statements: if \(f(x)=\begin{cases}x & 0 \leq x < 1 \\ 3-x & 1 \leq x \leq 2\end{cases}\) then

(a.) \(\lim_{x\rightarrow 1^-} f(x) = 1\)
(b.) \(\lim_{x\rightarrow 1^+} f(x) = 2\)
(c.) \(\lim_{x\rightarrow 1} f(x) = 2\)
(d.) None of the above

24. \(\lim _{x \rightarrow 0} \frac{\sin (a x)}{\sin (b x)}(a \neq 0, b \neq 0)\) is

(a.) \(\frac{a}{b}\)
(b.) \(\frac{b}{a}\)
(c.) \(a b\)
(d.) does not exist

25. What is the limit of \(\frac{ax + b}{cx}\) as \(x\) approaches infinity?

(a.) \(a/c\)
(b.) \(\infty\)
(c.) \(b\)
(d.) None of the above

26. The correct value of \(\lim_{x\rightarrow 0} \frac{x}{\sqrt{1-\cos x}}\) is

(a.) Does not exist
(b.) \(\sqrt{2}\)
(c.) \(-\sqrt{2}\)
(d.) \(\frac{1}{\sqrt{2}}\)

27. The limit \(\lim_{x\rightarrow 0^+} \frac{9}{x}\left(\frac{1}{\tan^{-1} x}-\frac{1}{x}\right)\) is

(a.) Does not exist
(b.) 9
(c.) 0
(d.) 3

28. If \(X\) and \(Y\) are two non-empty finite sets and \(f: X \rightarrow Y\) and \(g: Y \rightarrow X\) are mappings such that \(g \circ f: X \rightarrow X\) is a surjective (i.e., onto) map, then

(a.) \(f\) must be one-to-one
(b.) \(f\) must be onto
(c.) \(g\) must be one-to-one
(d.) \(X\) and \(Y\) must have the same number of elements

29. Let \(P(x)\) be a non-constant polynomial such that \(P(n)=P(-n)\) for all \(n \in \mathbb{N}\). Then \(P'(0)\) is

(a.) 1
(b.) 0
(c.) -1
(d.) Cannot be determined from the given data

30. If \(\lim_{x\rightarrow 0} \left(\frac{1+cx}{1-cx}\right)^{\frac{1}{x}} = 4\), then \(\lim_{x\rightarrow 0} \left(\frac{1+2cx}{1-2cx}\right)^{\frac{1}{x}}\) is

(a.) 2
(b.) 4
(c.) 16
(d.) 64