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

31. Let \(f(x)=e^{e^{-x}}\) and define \(g(x)=f(x+1)-f(x)\). Then, as \(x \rightarrow \infty\), the function \(g(x)\) converges to

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

32. Consider the functions \(f, g: \mathbb{Z} \rightarrow \mathbb{Z}\) defined by \(f(n)=3n+2\) and \(g(n)=n^{2}-5\).

(a.) Neither \(f\) nor \(g\) is a one-to-one function
(b.) The function \(f\) is one-to-one, but not \(g\)
(c.) The function \(g\) is one-to-one, but not \(f\)
(d.) Both \(f\) and \(g\) are one-to-one functions

33. Let \(T\) be the graph of the function

\[ f(x) = \begin{cases} 1+x, & -1 \leq x \leq 0 \\ 1-x, & 0 \leq x \leq 1 \end{cases} \]

Then the reflection of \(T\) in the line \(y=0\) is given by the graph of \(g(x)\) where

(a.) \(g(x)= \begin{cases}-1-x, & -1 \leq x \leq 0 \\ -1+x, & 0 \leq x \leq 1\end{cases}\)
(b.) \(g(x)= \begin{cases}-1+x, & -1 \leq x \leq 0 \\ -1-x, & 0 \leq x \leq 1\end{cases}\)
(c.) \(g(x)=\begin{cases}-1-x, & -1 \leq x \leq 0 \\ 1-x, & 0 \leq x \leq 1\end{cases}\)
(d.) \(g(x)=\begin{cases}1-x, & -1 \leq x \leq 0 \\ -1-x, & 0 \leq x \leq 1\end{cases}\)

34. Let \(a, b, c, d\) be rational numbers with \(ad-bc \neq 0\). Then the function \(f: \mathbb{R} \setminus \mathbb{Q} \rightarrow \mathbb{R}\) defined by \(f(x)=\frac{ax+b}{cx+d}\) is

(a.) Onto but not one-to-one.
(b.) One-to-one but not onto.
(c.) Neither one-to-one nor onto.
(d.) Both one-to-one and onto.

35. Let \(f:(0, \infty) \rightarrow \mathbb{R}\) be the function defined by \(f(x)=\frac{e^{x}}{x^{x}}\). Then \(\lim_{x\rightarrow \infty} f(x)\) is

(a.) Does not exist.
(b.) Exists and is 0.
(c.) Exists and is 1.
(d.) Exists and is \(e\).

36. \(\lim_{n\rightarrow \infty} \left(1-\frac{1}{n^2}\right)^n\) equals

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

37. \(\lim_{n\rightarrow \infty} \frac{1}{n^4} \sum_{j=0}^{2n-1} j^3\) equals

(a.) 4
(b.) 16
(c.) 1
(d.) 8

38. Which of the following are true for the function \(f(x)=\sin(x) \sin\left(\frac{1}{x}\right)\) with \(x \in (0,1)\)?

(a.) \(\varliminf_{x\rightarrow 0} f(x) = \varlimsup_{x\rightarrow 0} f(x)\)
(b.) \(\varliminf_{x\rightarrow 0} f(x) < \varlimsup_{x\rightarrow 0} f(x)\)
(c.) \(\varliminf_{x\rightarrow 0} f(x) = 1\)
(d.) \(\varlimsup_{x\rightarrow 0} f(x) = 0\)

39. The number of distinct real roots of the equation \(x^9+x^7+x^5+x^3+x+1=0\) is

(a.) 1
(b.) 3
(c.) 5
(d.) 9

40. The equation \(11^x+13^x+17^x-19^x=0\) has

(a.) No real root
(b.) Only one real root
(c.) Exactly two real roots
(d.) More than two real roots

41. Let \(f(x)=\tan^{-1} x\) for \(x \in \mathbb{R}\). Then

(a.) There exists a polynomial \(p(x)\) satisfying \(p(x) f'(x)=1\) for all \(x\)
(b.) \(f^{(n)}(0)=0\) for all positive even integers \(n\)
(c.) The sequence \(\{f^{(n)}(0)\}\) is unbounded
(d.) \(f^{(n)}(0)=0\) for all \(n\)