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WBJEE Limits and Continuity — practice questions

Explore 131 free Limits and Continuity questions in WBJEE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Limits and Continuity questions and exercises with answers

Showing questions 1–30 of 131.

Evaluate $\displaystyle\lim_{x \to 0} \dfrac{\sin 3x}{x}$.Read the answer and worked solution →Evaluate $\displaystyle\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}$.Read the answer and worked solution →Evaluate $\displaystyle\lim_{x \to 0} \dfrac{1 - \cos 2x}{x^2}$.Read the answer and worked solution →Evaluate $\displaystyle\lim_{x \to \infty} \left(1 + \dfrac{2}{x}\right)^x$.Read the answer and worked solution →Consider $f(x) = |x|$ at $x = 0$. Which statement is correct?Read the answer and worked solution →$\sin(180° - \theta)$ equals:Read the answer and worked solution →$\sin^2\theta + \cos^2\theta$ equals:Read the answer and worked solution →$\sin(A + B) = $Read the answer and worked solution →A **random variable** is best defined as:Read the answer and worked solution →A **discrete** random variable is one whose possible values:Read the answer and worked solution →For a continuous random variable $X$, the probability that $X$ equals any specific value is:Read the answer and worked solution →The **cumulative distribution function (c.d.f.)** $F(x)$ of a discrete random variable is:Read the answer and worked solution →When two fair dice are rolled and $X$ denotes the sum of the upper faces, the range of $X$ is:Read the answer and worked solution →If a fair coin is tossed twice and $X$ = number of heads, then $P[X = 1]$ equals:Read the answer and worked solution →If two cards are drawn with replacement from a deck of 52, $P[X = 2]$ where $X$ = number of aces equals:Read the answer and worked solution →A fair die is thrown. Let $X$ = number of factors of the number on the upper face. $P[X = 2]$ equals:Read the answer and worked solution →$f(x) = 3x^2$ for $0 < x < 1$ is the p.d.f. of $X$. The probability $P[1/2 < X < 1]$ is:Read the answer and worked solution →If $f(x) = kx^2(1 - x)$ for $0 < x < 1$ is the p.d.f. of $X$, the value of $k$ is:Read the answer and worked solution →For p.d.f. $f(x) = x/8$, $0 < x < 4$, the probability $P[x < 1.5]$ is:Read the answer and worked solution →Find $k$ if $f(x) = kx$ for $0 < x < 2$ and 0 otherwise is a valid p.d.f.Read the answer and worked solution →The c.d.f. $F(x)$ of the p.d.f. $f(x) = 3x^2$ for $0 < x < 1$ is:Read the answer and worked solution →Two cards are drawn with replacement. $X$ = number of aces. The probability $P[X \geq 1]$ equals:Read the answer and worked solution →For p.d.f. $f(x) = x^2/3$ for $-1 < x < 2$, the probability $P[0 < X \leq 1]$ equals:Read the answer and worked solution →For p.d.f. $f(x) = k(4 - x^2)$ for $-2 \leq x \leq 2$ (and 0 elsewhere), the value of $k$ is:Read the answer and worked solution →For the c.d.f. $F(x) = 3x - 2x^3$ (from p.d.f. $f(x) = 3(1 - 2x^2)$ on $[0, 1]$), $F(0.5)$ equals:Read the answer and worked solution →If a fair coin is tossed $n$ times, the probability of any exact pattern (like all heads) is:Read the answer and worked solution →If a die is thrown and 'getting an odd number' is success, then $p$ and $q$ are:Read the answer and worked solution →A bag has balls marked 0–9. Four balls drawn with replacement. Probability that **none** is marked '0' is:Read the answer and worked solution →Probability of bomb hitting target = 0.8. Out of 10 bombs dropped, probability that **exactly 2 miss** equals (with $p = 0.2$ miss):Read the answer and worked solution →sin(0°) equals:Read the answer and worked solution →

Worked practice questions

Evaluate $\displaystyle\lim_{x \to 0} \dfrac{\sin 3x}{x}$.

  1. $1$
  2. $3$
  3. $\infty$
  4. $0$
Show answer and solution

Correct answer: B. $3$

Using the standard limit $\lim_{u\to 0} \dfrac{\sin u}{u} = 1$, write $\dfrac{\sin 3x}{x} = 3 \cdot \dfrac{\sin 3x}{3x}$. As $x \to 0$, the second factor $\to 1$, giving $3$.

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Evaluate $\displaystyle\lim_{x \to 2} \dfrac{x^2 - 4}{x - 2}$.

  1. The limit does not exist
  2. $4$
  3. $2$
  4. $0$
Show answer and solution

Correct answer: B. $4$

Factor the numerator: $x^2 - 4 = (x-2)(x+2)$. Cancelling, $\dfrac{(x-2)(x+2)}{x-2} = x + 2$ for $x \neq 2$. Limit = $4$. Option D is the trap if you stop at the $0/0$ form without factoring.

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Evaluate $\displaystyle\lim_{x \to 0} \dfrac{1 - \cos 2x}{x^2}$.

  1. $2$
  2. $4$
  3. $0$
  4. $1$
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Correct answer: A. $2$

Use $1 - \cos 2x = 2\sin^2 x$. Then $\dfrac{2\sin^2 x}{x^2} = 2\left(\dfrac{\sin x}{x}\right)^2 \to 2 \cdot 1 = 2$ as $x \to 0$. Alternatively, expand $\cos 2x \approx 1 - 2x^2$ to leading order, giving $\dfrac{2x^2}{x^2} = 2$.

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More Mathematics topics

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