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Statistics · Class 12 Commerce

Ch 8Limit — Class 12 Statistics, concept-first.

The idea of a limit is the foundation on which the whole of calculus — and every business-mathematics application built on it, from continuous compounding to marginal analysis in the very next chapter — rests.

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Key concepts

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Standard Limits

Some limits recur so often across problems that it is worth memorising their values outright, along with the one theorem that proves the trickiest of them: the Sandwich (Squeeze) Theorem.

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Meaning and Notation of Limit

The idea of a limit is the foundation on which the whole of calculus — and every business-mathematics application built on it, from continuous compounding to marginal analysis in the very next chapter…

2

Evaluating Limits: Algebra of Limits and Indeterminate Forms

Once the meaning of a limit is clear, the practical skill this chapter builds is evaluating limits without drawing a graph or plugging in a long table of values every time.

3

Standard Limits

Rather than factorising or rationalising from scratch every time, certain limit forms recur so often — in this Gujarat Std-12 Statistics chapter and in the differentiation chapter that follows it — th…

4

Continuity of a Function at a Point

A function's continuity at a point is a direct application of the LHL = RHL idea from Section 1, with one extra condition added: the common limit value must also match the function's actual value at t…

5

Business Applications of Limits

Limits are not merely an algebraic exercise in the Gujarat Std-12 Statistics (Business Mathematics Statistics) syllabus — they underpin three genuinely practical results a commerce student meets aga…

Exercises

+Show 8 questions8 questions
  1. Q6Evaluate: $\displaystyle\lim_{x\to 3}\dfrac{x^{3}-27}{x-3}$Free
  2. Q7Evaluate: $\displaystyle\lim_{x\to 1}\dfrac{x^{2}-1}{x^{3}-1}$Free
  3. Q8Given $f(x)=\begin{cases}2x+1, & x<2\\ x^{2}-1, & x\ge 2\end{cases}$, find the left-hand limit and the right-hand limit of $f(x)$ at $x=2$,…Free
  4. Q9Examine the continuity of $f(x)=\begin{cases}x+2, & x<1\\ 3, & x=1\\ x^{2}+1, & x>1\end{cases}$ at $x=1$.Preview
  5. Q10Evaluate: $\displaystyle\lim_{x\to 0}\dfrac{5^{x}-1}{x}$Preview
  6. Q11Which of the following equals $\displaystyle\lim_{x\to 0}\dfrac{e^{x}-1}{x}$? (a) 0 (b) 1 (c) $e$ (d) does not existPreview
  7. Q12A firm's total fixed cost is Rs. 50,000 per month. Its average fixed cost at output $x$ units is $\text{AFC}(x)=\dfrac{50{,}000}{x}$. Find $…Preview
  8. Q13A wholesaler's total billed cost for $x$ units of a commodity is $C(x)=100x$ for $0<x\le50$, and $C(x)=90x$ for $x>50$ (the lower rate appli…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 27 questions27 questions
  1. Q1What is the value of $\lim_{x \to 3} \, 3x - 1$? (a) $9$ (b) $10$ (c) $\frac{4}{3}$ (d) $8$Preview
  2. Q2Explain the meaning of $x \to 0$.Preview
  3. Q3Express $|x + 1| < 0.5$ in neighbourhood and interval form. OR State multiplication and division working rule of limit.Preview
  4. Q4Find the value of $\lim_{x \to 3} \dfrac{x^2 - 2x - 3}{x^2 - 5x + 6}$Preview
  5. Q5Find the value of $\lim_{x \to 1} \dfrac{\sqrt{x+3} - 2}{\sqrt{x+8} - 3}$Preview
  6. Q6What is the value of $\lim_{x \to -2} 10$ ? (a) 10 (b) $-2$ (c) 8 (d) IndeterminatePreview
  7. Q7What is the value of $\lim_{x \to 3} \dfrac{x^4 - 81}{x - 3}$. (a) 192 (b) 324 (c) 36 (d) 108Preview
  8. Q8Define the $\delta$ neighbourhood of $a$.Preview
  9. Q9Express $|x + 1| < 0.5$ in neighbourhood and interval form.Preview
  10. Q10Explain the meaning of $x \to 0$.Preview
  11. Q11Find the value of $\lim_{x \to -3} \dfrac{2x^2 + 7x + 3}{3x^2 + 8x - 3}$.Preview
  12. Q12What is the value of $\lim_{x \to 3} (3x - 1)$? (a) (A) $9$ (b) (B) $10$ (c) (C) $\frac{4}{3}$ (d) (D) $8$Preview
  13. Q13Express $0.001$ neighbourhood of $-5$ in modulus form.Preview
  14. Q14If $|x - 10| < k_1 = (k_2, 10.01)$, then find the values of $k_1$ and $k_2$. OR Express $|x + 1| < 0.5$ in neighborhood and interval form.Preview
  15. Q15Find the value of $\lim_{x \to 3} \frac{x^5 - 243}{x - 3}$.Preview
  16. Q16Find the value of $\lim_{x \to -3} \frac{2x^2 + 7x + 3}{3x^2 + 8x - 3}$.Preview
  17. Q17What is the neighbourhood form of $|x - 5| < 0.25$ ? (a) $N(0.25, 5)$ (b) $N(-5, 0.25)$ (c) $N(-5, -0.25)$ (d) $N(5, 0.25)$Preview
  18. Q18If $y = 10 - 3x$ and $x \to -3$ then $y$ tends to which value? (a) $1$ (b) $9$ (c) $19$ (d) $7$Preview
  19. Q19If $\lim_{x \to -1} 4x + k = 6$ then find the value of $k$.Preview
  20. Q20Express $N(16, 0.5)$ in the interval and modulus form.Preview
  21. Q21Find the value of $\lim_{x \to -2} \frac{x^7 + 128}{x + 2}$Preview
  22. Q22Find the value of $\lim_{x \to 2} \frac{f(x) - f(2)}{x - 2}$ where $f(x) = x^2 + x$.Preview
  23. Q23What is the modulus form of 0.3 neighbourhood of 3? (a) $|x - 0.3| < 3$ (b) $|x - 3| < 0.3$ (c) $|x + 3| < 0.3$ (d) $|x - 3| > 0.3$Preview
  24. Q24What is the value of $\lim_{x \to 4} \sqrt{4x + 9}$? (a) $5$ (b) $25$ (c) $\dfrac{7}{4}$ (d) $7$Preview
  25. Q25Express $|x - 10| < \dfrac{1}{10}$ in neighbourhood form.Preview
  26. Q26If $N(3, b) = (2.95, k)$ then find the values of $b$ and $k$.Preview
  27. Q27Find the value of $\lim_{x \to 1} \dfrac{3x^2 - 4x + 1}{x - 1}$.Preview

More questions

+Show 5 questions5 questions
  1. Example 1Evaluate: $\displaystyle\lim_{x\to 2}\dfrac{x^{2}-4}{x-2}$Free
  2. Example 2Evaluate: $\displaystyle\lim_{x\to 0}\dfrac{e^{3x}-1}{x}$Free
  3. Example 3Evaluate: $\displaystyle\lim_{x\to \infty}\dfrac{3x^{2}+2x-1}{5x^{2}-x+4}$Preview
  4. Example 4Evaluate: $\displaystyle\lim_{x\to 0}\dfrac{\log_{e}(1+5x)}{x}$Preview
  5. Example 5A sum of Rs. 10,000 is invested at a nominal annual interest rate of 8%, compounded continuously. (i) Derive the continuous-compounding form…Preview