Academia Aeternum
तमसो मा ज्योतिर्गमय
Wisdom · Knowledge · Eternity
Where Calculus Begins — Every Limit & Derivative Exercise Solved
3 exercise files · 73 total questions
\(\lim_{x\to a}\dfrac{x^n-a^n}{x-a} = na^{n-1}\)\(\lim_{x\to 0}\dfrac{\sin x}{x} = 1;\quad \lim_{x\to 0}\dfrac{\tan x}{x} = 1\)\(\dfrac{d}{dx}(x^n)=nx^{n-1};\quad \dfrac{d}{dx}(\sin x)=\cos x\)\((uv)'=u'v+uv';\quad \left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}\)Step 1 — Try direct substitution first. If 0/0 or ∞/∞, proceed. Step 2 — Algebraic 0/0: factorise; cancel (x−a); substitute. Step 3 — Trig limits: manipulate to get sinx/x form using ×÷ and substitution. Step 4 — First principles: write f(x+h)−f(x); expand; cancel h; take h→0. Step 5 — Product/quotient rule: label u and v clearly first.
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