Business Mathematics and Basic Statistics · Class 11 Commerce
Ch 10Limits and Derivatives — Class 11 Business Mathematics and Basic Statistics, concept-first.
When we study how a function behaves as gets closer and closer to some fixed value — without necessarily ever reaching itself — we are studying the limit of as approaches . The formal notation for this is
Key concepts
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Left-Hand Limit and Right-Hand Limit
The LHL and RHL are the values approaches from below and above respectively. The two-sided limit exists only when LHL RHL; this is checked piece-by-piece for a piecewise-defined function at its breakpoint.
Most relevant Q&A
In previous exams
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Chapter contents
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The Idea of a Limit and Limit Notation
When we study how a function behaves as gets closer and closer to some fixed value — without necessarily ever reaching itself — we are studying the limit of as approaches .
Left-Hand Limit and Right-Hand Limit
Because can approach from two directions — from values less than or from values greater than — a limit is more precisely built from two one-sided limits:
Algebra of Limits: Sum, Difference, Product and Quotient
If and both exist, the following rules — together called the algebra of limits — let us break a complicated limit into simpler pieces:
The Standard Algebraic Limit Formula and Its Equivalent Form
Many limits that look like on direct substitution — because both the numerator and denominator vanish at — are handled by one standard result:
Exponential and Logarithmic Standard Limits
Two more standard limits — both again of the indeterminate form on direct substitution — are used constantly once exponential and logarithmic functions enter business-mathematics problems (compound gr…
The Derivative as a Rate of Measure
The derivative of a function , written or , measures the instantaneous rate at which changes with respect to at a given point — how fast the output is changing per unit change in the input, at that ex…
Differentiation of Standard Functions
The following standard differentiation formulae are used directly, as given rules, throughout this chapter:
The Chain Rule
None of §7's five formulae, on their own, can differentiate a composite function — a function of a function, such as or , where the "outer" function ( or ) is applied not to directly but to another fu…
Application: Maxima and Minima for Cost, Demand and Marginal-Cost Functions
Differentiation has a direct business use: finding the output level at which a cost function is smallest, or a revenue/profit function is largest.
More questions
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- Example 1A mobile-recharge plan charges $f(x) = 2x$ (rupees) for $x$ minutes of talk-time when $x \le 10$, and $f(x) = 3x - 10$ (rupees) when $x > 10…Free
- Example 2A sales agent earns commission $g(x) = 500$ (a flat amount, in rupees) for sales $x \le 50{,}000$, and $g(x) = 0.02x$ for sales $x > 50{,}00…Free
- Example 3Evaluate $\displaystyle\lim_{x\to2}\big(4x^3-2x^2+7\big)$ using the algebra of limits.Free
- Example 4Evaluate $\displaystyle\lim_{x\to2}\frac{x^2+3x-4}{x^3-1}$ using the algebra of limits.Preview
- Example 5Evaluate $\displaystyle\lim_{x\to3}\frac{x^5-243}{x-3}$ using the standard algebraic limit formula.Preview
- Example 6Evaluate $\displaystyle\lim_{x\to4}\frac{\sqrt{x}-2}{x-4}$ using the standard algebraic limit formula.Preview
- Example 7Show that $\displaystyle\lim_{x\to0}\frac{(1+x)^3-1}{x}$ (evaluated using the special formula) gives the same value as $\displaystyle\lim_{x…Preview
- Example 8Evaluate $\displaystyle\lim_{x\to0}\frac{e^{5x}-1}{x}$.Preview
- Example 9Evaluate $\displaystyle\lim_{x\to0}\frac{\ln(1+7x)}{x}$.Preview
- Example 10Evaluate $\displaystyle\lim_{x\to0}\frac{(e^{2x}-1)-\ln(1+3x)}{x}$.Preview
- Example 11Differentiate $y = 6x^{5} - 4e^{x} + 3\ln x - 2(5^{x}) + 11$ with respect to $x$.Preview
- Example 12Differentiate with respect to $x$: (i) $y = e^{3x^{2}-1}$ (ii) $y = \ln(4x^{3}+2x)$.Preview
- Example 13The total cost (in rupees) of producing $x$ units of a good is $C(x) = x^{3} - 15x^{2} + 100x + 200$. Find the marginal cost function $MC(x)…Preview
- Example 14A firm's demand function is $p = 200 - 4x$ (price $p$ in rupees when $x$ units are demanded), so the revenue function is $R(x) = p\cdot x =…Preview