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Business Mathematics and Statistics · Class 12 Commerce

Ch 5Limit and Continuity — Class 12 Business Mathematics and Statistics, concept-first.

Business decisions are full of questions about tendency rather than an exact value: as the number of units sold gets closer and closer to a factory's full capacity, what does the average cost per unit approach? As a loan's repayment period stretches out further and further, what does the total interest paid tend toward…

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Key concepts

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Indeterminate Forms

An indeterminate form such as or , produced when direct substitution is attempted, means only that direct substitution has failed to answer the question — not that the limit fails to exist or automatically equals zero or…

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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 Intuitive Idea of a Limit

Business decisions are full of questions about tendency rather than an exact value: as the number of units sold gets closer and closer to a factory's full capacity, what does the average cost per unit…

2

Left-Hand Limit, Right-Hand Limit and Existence of a Limit

A value of can approach a point from two directions — from values less than (from the left) or from values greater than (from the right) — and a function's tendency can genuinely differ depending on w…

3

Algebra of Limits and Methods of Evaluation

Once the existence of a limit is settled, evaluating it is made far easier by a small set of algebraic rules. If and (both limits existing), then:

4

Standard Limits

Certain limits recur so often in business mathematics — particularly once differentiation is introduced in the next chapter — that it is worth learning their values as standard results, rather than re…

5

Indeterminate Forms

When direct substitution is tried on a limit and produces an expression like or , the result is called an indeterminate form — it does not mean the limit fails to exist, and it certainly does not mean…

6

Continuity of a Function

A function's graph that can be drawn without ever lifting the pen from the paper is, informally, a continuous function — but business mathematics needs a precise test, not just a picture, since a func…

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Business Applications: Continuous Compounding and the Continuity of Cost Functions

Limits and continuity are not confined to abstract algebra — two of the most practical ideas in business mathematics are built directly on them, and both are regularly tested in Odisha CHSE Business M…

Exercises

+Show 7 questions7 questions
  1. Q8Evaluate $\lim_{x \to 1} \dfrac{x^3-1}{x^2-1}$.Free
  2. Q9Evaluate $\lim_{x \to \infty} \dfrac{3x^2+5x-1}{5x^2-2x+7}$.Free
  3. Q10Choose the correct value of $\lim_{x \to 0} \dfrac{\sin x}{x}$ (with $x$ in radians): (a) $0$ (b) $1$ (c) $\infty$ (d) does not existFree
  4. Q11For a function $f$ to be continuous at $x=a$, which of the following conditions must ALL hold? (a) $f(a)$ is defined, and nothing else (b) $…Preview
  5. Q12Check the continuity of the function $$f(x) = \begin{cases} x+2, & x<1 \\ 5, & x=1 \\ 3x, & x>1 \end{cases}$$ at $x=1$.Preview
  6. Q13A wholesaler's pricing policy is: for an order of $x$ units, the total cost is $C(x) = 50x$ rupees if $x \leq 100$, and $C(x) = 45x$ rupees…Preview
  7. Q14Using the standard limit $\lim_{n \to \infty}\left(1+\frac1n\right)^n = e$, derive the formula for the amount under continuous compounding,…Preview

Sample & Board Papers

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

+Show 11 questions11 questions
  1. Q1Answer each of the following questions in one sentence: (x) Evaluate: $\lim\limits_{x \to 2} \dfrac{x^2 + 4}{x + 2}$.Preview
  2. Q2(d) $\lim\limits_{x \to 2} \dfrac{x^2 - 4}{x - 2}$ is equal to (a) $4$ (b) $2$ (c) $0$ (d) $\infty$Preview
  3. Q3(c) Test the continuity of the following function at $x = 1$: $f(x) = \begin{cases} \dfrac{x^2 - 4x + 3}{x - 1}, & x \neq 1 \\ -2, & x = 1 \…Preview
  4. Q4Evaluate: $\lim\limits_{x \to 2} \left[ \dfrac{1}{x - 2} - \dfrac{2(2x - 3)}{x^3 - 3x^2 + 2x} \right]$Preview
  5. Q5$\lim\limits_{x \to a} \dfrac{x^n - a^n}{x - a}$ is equal to : (a) $x^n - a^n$ (b) $an^{-1}$ (c) $na$ (d) $na^{n-1}$Preview
  6. Q6Answer the following questions within one sentence each : Evaluate : $\lim\limits_{x \to 3}(x^2 + 3)$Preview
  7. Q7Find the value of $\lim\limits_{x \to 5} \dfrac{x^2 - 25}{x - 5}$Preview
  8. Q8Write the conditions to be satisfied by a function to be continuous at $x = a$.Preview
  9. Q9The value of $\lim_{x \to 3} \frac{x^2+9}{x+3}$ is : (a) 1 (b) 2 (c) 3 (d) 4Preview
  10. Q10Evaluate $\lim_{x \to 2} \frac{x^2-4}{x-2}$.Preview
  11. Q11Evaluate : $\lim_{x \to 1} \frac{x^2+4x-5}{x-1}$Preview

More questions

+Show 7 questions7 questions
  1. Example 1A function $f(x)$ is defined as $f(x) = x+1$ for $x<2$, and $f(x) = x^2$ for $x \geq 2$. Find the left-hand limit and the right-hand limit o…Free
  2. Example 2Evaluate $\lim_{x \to 3} (2x^2 - 5x + 1)$.Free
  3. Example 3Evaluate $\lim_{x \to 3} \dfrac{x^3-27}{x-3}$.Free
  4. Example 4Evaluate $\lim_{x \to 4} \dfrac{\sqrt{x}-2}{x-4}$.Preview
  5. Example 5Evaluate $\lim_{x \to 0} \dfrac{\sin 5x}{x}$.Preview
  6. Example 6Show, using a table of values, that $\lim_{n \to \infty} \left(1+\dfrac1n\right)^n = e \approx 2.71828$.Preview
  7. Example 7Evaluate $\lim_{x \to 0} \dfrac{e^{3x}-1}{x}$.Preview