Limits and Derivatives
Ye chapter calculus ka darwaaza hai. "Limit" batata hai ki kisi function ki value kahaan ja rahi hai jab input kisi point ke paas pahunchta hai. "Derivative" batata hai ki function kitni tezi se badal raha hai (rate of change / slope).
1. Limit ka Idea
\(\displaystyle\lim_{x \to a} f(x) = L\) ka matlab — jaise-jaise \(x\), \(a\) ke kareeb jaata hai, \(f(x)\) value \(L\) ke kareeb jaati hai. (\(x = a\) par function ki value se koi farak nahi padta.)
Left aur Right Hand Limits
- LHL: \(\displaystyle\lim_{x \to a^{-}} f(x)\) — \(x\), \(a\) se chhoti taraf se aaye.
- RHL: \(\displaystyle\lim_{x \to a^{+}} f(x)\) — \(x\), \(a\) se badi taraf se aaye.
Limit tabhi exist karti hai jab LHL \(=\) RHL.
2. Algebra of Limits
Agar \(\lim f(x)\) aur \(\lim g(x)\) dono exist karte hain:
- \(\lim [f(x) \pm g(x)] = \lim f(x) \pm \lim g(x)\)
- \(\lim [f(x)\,g(x)] = \lim f(x) \cdot \lim g(x)\)
- \(\lim \dfrac{f(x)}{g(x)} = \dfrac{\lim f(x)}{\lim g(x)}\), jab \(\lim g(x) \neq 0\)
3. Kuch Standard Limits
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
\[ \lim_{x \to 0} \frac{\tan x}{x} = 1 \]
\[ \lim_{x \to a} \frac{x^{n} - a^{n}}{x - a} = n\,a^{\,n-1} \]
\[ \lim_{x \to 0} \frac{a^{x} - 1}{x} = \log_e a, \qquad \lim_{x \to 0} \frac{e^{x} - 1}{x} = 1 \]
\(\displaystyle\lim_{x \to 3} (x^{2} + 2x - 1)\). Yahan seedha substitute karo: \[ = 3^{2} + 2(3) - 1 = 9 + 6 - 1 = 14. \]
\(\displaystyle\lim_{x \to 2} \frac{x^{2} - 4}{x - 2}\). Direct daalne par \(\frac{0}{0}\) (indeterminate), isliye factor karo: \[ = \lim_{x \to 2} \frac{(x-2)(x+2)}{x - 2} = \lim_{x \to 2} (x + 2) = 4. \]
4. Derivative — Definition
Function \(f\) ka derivative \(x\) par: \[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}, \] agar ye limit exist kare. Ye \(x\) par curve ki slope hai.
Notation: \(f'(x)\), \(\dfrac{dy}{dx}\), ya \(\dfrac{d}{dx}f(x)\) — sab same.
\(f(x) = x^{2}\) ka derivative: \[ f'(x) = \lim_{h \to 0} \frac{(x+h)^{2} - x^{2}}{h} = \lim_{h \to 0} \frac{2xh + h^{2}}{h} = \lim_{h \to 0}(2x + h) = 2x. \]
5. Rules of Differentiation
- Power rule: \(\dfrac{d}{dx}(x^{n}) = n\,x^{\,n-1}\)
- Constant: \(\dfrac{d}{dx}(c) = 0\)
- Sum/Difference: \((f \pm g)' = f' \pm g'\)
- Product rule: \((fg)' = f'g + fg'\)
- Quotient rule: \(\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^{2}}\)
Common derivatives
\[ \frac{d}{dx}(\sin x) = \cos x, \qquad \frac{d}{dx}(\cos x) = -\sin x \]
\[ \frac{d}{dx}(e^{x}) = e^{x}, \qquad \frac{d}{dx}(\log_e x) = \frac{1}{x} \]
\(y = x^{3} + 5x^{2} - 2x + 7\) ka derivative: \[ \frac{dy}{dx} = 3x^{2} + 10x - 2. \]
Key Takeaways
- Limit exist karti hai jab LHL \(=\) RHL.
- \(\frac{0}{0}\) aaye to factor/simplify karke limit nikaalo.
- Yaad rakho: \(\lim_{x\to 0}\frac{\sin x}{x} = 1\) aur \(\lim_{x\to a}\frac{x^{n}-a^{n}}{x-a} = na^{n-1}\).
- Derivative \(= \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\) = slope.
- Power, sum, product, quotient rules differentiation ki base hain.