Calculus Foundations and Problem Solving - BSEB Inter Study Guide
Calculus Foundations and Problem Solving
This domain ties together limits, standard derivatives and integrals, and the algebraic manipulation needed to start a calculus problem. BSEB papers reward recognising a standard form before differentiating or integrating.
Core rules
- lim x→a f(x) exists only if left and right limits agree.
- Standard limits: lim x→0 (sin x)/x=1; lim x→0 (1−cos x)/x²=1/2; lim x→0 (e^x−1)/x=1; lim x→0 ln(1+x)/x=1.
- For 0/0 or ∞/∞ forms, factorise, rationalise or use the standard limit before anything else.
- d/dx(x^n)=nx^(n−1); d/dx(e^x)=e^x; d/dx(ln x)=1/x.
- ∫x^n dx=x^(n+1)/(n+1)+C for n not equal to −1; ∫e^x dx=e^x+C.
Worked example
Evaluate lim x→0 (sin 3x)/x. Write (sin 3x)/x=3·(sin 3x)/(3x). As x→0, 3x→0, so (sin 3x)/(3x)→1. Hence the limit is 3·1=3.
Traps
- Substituting x=0 and writing 0/0 as the answer.
- Cancelling terms inside a limit without checking they are nonzero.
- Forgetting the chain rule when a standard derivative has an inner function.
- Mixing derivative and integral formulas, especially the sign in ∫sin x dx.
- Applying a standard limit when the argument does not tend to zero.