Q1. Evaluate lim(x→2) (x² − 4)/(x − 2).
Factor the numerator: x² − 4 = (x−2)(x+2)
(x² − 4)/(x − 2) = (x−2)(x+2)/(x−2) = x + 2 (for x ≠ 2)
lim(x→2) (x+2) = 2 + 2 = 4
Limits and Derivatives — Class 11
Calculus begins here. A limit asks what a function approaches; a derivative asks how fast it's changing. Both start with simple rules.
Limitlim(x→a) f(x) is the value f(x) approaches as x gets arbitrarily close to a (without necessarily reaching it).
Derivativef'(x) = lim(h→0) [f(x+h) − f(x)]/h — the instantaneous rate of change of f at x.
Power ruled/dx(xⁿ) = n·xⁿ⁻¹ — the single most-used derivative rule.
Solved examples
Q2. Differentiate f(x) = 3x⁴ − 2x + 5.
Apply the power rule term by term: d/dx(3x⁴) = 12x³
d/dx(−2x) = −2, d/dx(5) = 0
f'(x) = 12x³ − 2
Formula sheet
| Quantity | Formula |
|---|---|
| Standard limit | lim(x→0) (sinx)/x = 1 |
| Power rule | d/dx(xⁿ) = n xⁿ⁻¹ |
| Sum rule | (f+g)' = f' + g' |
| Constant rule | d/dx(c) = 0 |
Common mistakes & exam wins
- For a 0/0 limit, try factoring — a common factor usually cancels to reveal the answer.
- The derivative of a constant is always 0.
- Apply the power rule to EACH term separately when differentiating a polynomial.
Frequently asked questions
What is a limit in calculus?
The value a function approaches as the input gets arbitrarily close to some point, written lim(x→a) f(x).
What is the power rule for derivatives?
d/dx(xⁿ) = n·xⁿ⁻¹ — multiply by the exponent and reduce the exponent by 1.
What is the derivative of a constant?
Zero — a constant never changes, so its rate of change is 0.