Mathematics and Statistics · Class 11 Commerce
Ch 7Limits — Class 11 Mathematics and Statistics, concept-first.
A limit describes the single value that a function approaches as its input approaches some fixed number — without any concern for what happens exactly at . This distinction is the whole point of the idea: a limit asks about the neighbourhood around , never about the point itself.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
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…
Most relevant Q&A
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
The Concept of a Limit
A limit describes the single value that a function approaches as its input approaches some fixed number — without any concern for what happens exactly at .
Left-hand and Right-hand (One-sided) Limits
can approach from two directions, and the function may behave differently on each side. This gives two one-sided limits:
Algebra of Limits
Once individual limits are known, they combine by simple arithmetic rules. Suppose and , both existing (finite). Then:
Standard Limits
Certain limits recur so often — and cannot be found by mere substitution — that their results are memorised as standard limits. Each is stated below with the exact conditions under which it holds.
Limits at Infinity
Instead of approaching a finite number, we may ask what approaches as grows without bound — written (and similarly ).
Indeterminate Forms and How to Resolve Them
When direct substitution produces an expression with no determinable value on its own, the result is called an indeterminate form.
Exercises
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- Q11Evaluate $\displaystyle\lim_{x\to 1}\frac{x^{2} + 2x - 3}{x - 1}$.Free
- Q12Evaluate $\displaystyle\lim_{x\to 0}\frac{1 - \cos x}{x^{2}}$.Free
- Q13Evaluate $\displaystyle\lim_{x\to 0}\frac{3^{x} - 1}{x}$.Preview
- Q14Evaluate $\displaystyle\lim_{x\to\infty}\frac{2x + 5}{3x^{2} - x + 1}$.Preview
- Q15Evaluate $\displaystyle\lim_{x\to 2}\left(\frac{1}{x - 2} - \frac{4}{x^{2} - 4}\right)$.Preview
- Q16A function is defined by $f(x)=x+5$ for all $x\neq 1$, and $f(1)=20$. Find $\displaystyle\lim_{x\to 1} f(x)$, and state whether it equals $f…Preview
More questions
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- Example 1Evaluate $\displaystyle\lim_{x\to 3}\big(2x^2 - 5x + 1\big)$.Free
- Example 2Evaluate $\displaystyle\lim_{x\to 2}\frac{x^2 - 4}{x - 2}$.Free
- Example 3Evaluate $\displaystyle\lim_{x\to 4}\frac{x^{3} - 64}{x - 4}$.Free
- Example 4Evaluate $\displaystyle\lim_{x\to 0}\frac{\sin 5x}{3x}$.Preview
- Example 5Evaluate $\displaystyle\lim_{x\to 0}\frac{\tan 7x}{\sin 2x}$.Preview
- Example 6Evaluate $\displaystyle\lim_{x\to 0}\frac{e^{3x} - 1}{x}$.Preview
- Example 7Evaluate $\displaystyle\lim_{x\to 0}\frac{\log_e (1 + 4x)}{x}$.Preview
- Example 8Evaluate $\displaystyle\lim_{x\to 9}\frac{\sqrt{x} - 3}{x - 9}$.Preview
- Example 9Evaluate $\displaystyle\lim_{x\to\infty}\frac{5x^{2} - 3x + 2}{4x^{2} + x - 7}$.Preview
- Example 10Evaluate the one-sided limits and hence $\displaystyle\lim_{x\to 0}\frac{|x|}{x}$, if it exists.Preview