Real Analysis MCQs Math

\[
\textbf{MCQs on Real Analysis with Answers}
\]

\[
\textbf{Q1: Which of the following is a limit point of the set } \{1, 2, 3, 4, 5\} \text{?}
\]
\[
\text{(A) } 2 \quad
\text{(B) } 3 \quad
\text{(C) } 6 \quad
\text{(D) } None of these
\]
\[
\textbf{Answer: (D) } None of these
\]

\[
\textbf{Q2: The set of real numbers is:}
\]
\[
\text{(A) } Countable \quad
\text{(B) } Uncountable \quad
\text{(C) } Discrete \quad
\text{(D) } Finite
\]
\[
\textbf{Answer: (B) } Uncountable
\]

\[
\textbf{Q4: If } f(x) = x^2, \text{ what is } \lim_{x \to 3} f(x) \text{?}
\]
\[
\text{(A) } 6 \quad
\text{(B) } 9 \quad
\text{(C) } 3 \quad
\text{(D) } 0
\]
\[
\textbf{Answer: (B) } 9
\]

\[
\textbf{Q5: Which of the following is a Cauchy sequence?}
\]
\[
\text{(A) } \left\{ 1, 1/2, 1/3, \dots \right\} \quad
\text{(B) } \left\{ 1, 1/2, 1/4, 1/8, \dots \right\} \quad
\text{(C) } \left\{ 1, 2, 3, 4, \dots \right\} \quad
\text{(D) } None of these
\]
\[
\textbf{Answer: (B) } \left\{ 1, 1/2, 1/4, 1/8, \dots \right\}
\]

\[
\textbf{Q6: Which of the following is a convergent series?}
\]
\[
\text{(A) } \sum_{n=1}^{\infty} \frac{1}{n} \quad
\text{(B) } \sum_{n=1}^{\infty} \frac{1}{n^2} \quad
\text{(C) } \sum_{n=1}^{\infty} 2^n \quad
\text{(D) } None of these
\]
\[
\textbf{Answer: (B) } \sum_{n=1}^{\infty} \frac{1}{n^2}
\]

\[
\textbf{Q8: The derivative of the function } f(x) = x^3 + 5x^2 \text{ is:}
\]
\[
\text{(A) } 3x^2 + 5x \quad
\text{(B) } 3x^2 + 10x \quad
\text{(C) } 3x^2 + 2x \quad
\text{(D) } 2x^2 + 5x
\]
\[
\textbf{Answer: (B) } 3x^2 + 10x
\]

\[
\textbf{Q9: The function } f(x) = \frac{1}{x} \text{ has a discontinuity at:}
\]
\[
\text{(A) } x = 0 \quad
\text{(B) } x = 1 \quad
\text{(C) } x = 2 \quad
\text{(D) } None of these
\]
\[
\textbf{Answer: (A) } x = 0
\]

\[
\textbf{Q10: The set of all limit points of a set } A \text{ is called:}
\]
\[
\text{(A) } Closure of A \quad
\text{(B) } Interior of A \quad
\text{(C) } Boundary of A \quad
\text{(D) } None of these
\]
\[
\textbf{Answer: (A) } Closure of A
\]