| --- |
| source: ncert |
| topic: calculus |
| class_level: class_11 |
| chapter: ch13 |
| difficulty: intermediate |
| --- |
| |
| Class 11 | Ch13: Limits and Derivatives |
| LIMIT: lim(x→a) f(x) = L means f(x) approaches L as x approaches a. |
| Left limit: lim(x→a-) f(x). Right limit: lim(x→a+) f(x). |
| Limit exists iff left limit = right limit. |
| STANDARD LIMITS: |
| lim(x→0) sin(x)/x = 1. lim(x→0) (1-cos(x))/x = 0. |
| lim(x→0) tan(x)/x = 1. lim(x→0) (e^x-1)/x = 1. |
| lim(x→0) (a^x-1)/x = ln(a). lim(x→a) (x^n-a^n)/(x-a) = na^(n-1). |
| ALGEBRA OF LIMITS: |
| lim(f±g)=lim f ± lim g. lim(fg)=lim f × lim g. lim(f/g)=lim f/lim g (if lim g≠0). |
| lim(cf)=c×lim f. lim(f^n)=(lim f)^n. |
| DERIVATIVES (FIRST PRINCIPLES): |
| f'(x)=lim(h→0) [f(x+h)-f(x)]/h. |
| STANDARD DERIVATIVES: |
| d/dx[x^n]=nx^(n-1). d/dx[sin x]=cos x. d/dx[cos x]=-sin x. |
| d/dx[tan x]=sec^2 x. d/dx[e^x]=e^x. d/dx[ln x]=1/x. |
| d/dx[constant]=0. d/dx[cf(x)]=cf'(x). d/dx[f±g]=f'±g'. |
| PRODUCT RULE: d/dx[fg]=f'g+fg'. |
| QUOTIENT RULE: d/dx[f/g]=(f'g-fg')/g^2. |
| SOLVED EXAMPLES: |
| Example 1: lim(x→0) sin(3x)/x=3×lim(x→0) sin(3x)/(3x)=3×1=3. |
| Example 2: Derivative of x^3 sin x. |
| d/dx[x^3 sin x]=3x^2 sin x+x^3 cos x (product rule). |
| Example 3: Derivative of (x^2+1)/(x^2-1). |
| Using quotient rule: [(2x)(x^2-1)-(x^2+1)(2x)]/(x^2-1)^2. |
| =[2x^3-2x-2x^3-2x]/(x^2-1)^2=(-4x)/(x^2-1)^2. |
| Example 4: lim(x→2) (x^3-8)/(x-2)=lim(x→2) (x^2+2x+4)=4+4+4=12. |
| Using formula: na^(n-1)=3×2^2=12. Confirmed. |
| COMMON MISTAKES: |
| lim(x→0) sin(x)/x=1 only when angle is in RADIANS. |
| Product rule: d/dx[fg]=f'g+fg' NOT f'g'. |
| Quotient rule: numerator is f'g-fg' (not fg'-f'g). |
