Explanation

• L'Hospital's Rule is only used when you get 0/0 or ∞/∞ form after direct substitution.

Conditions for L'Hospital's Rule:

1. Indeterminate form → Must be 0/0 or ∞/∞ at the limit point. If you get 5/2 or 7, don’t use L'Hospital.
2. Both numerator + denominator must be differentiable near the point
3. Denominator’s derivative ≠ 0 near the point

So "indeterminate" is the key condition.

Last verified on 05-06-2026

Explanation

If aₙ means the coefficient of xⁿ in the power series/Taylor series of f(x) = x - x³, then:

f(x) = x - x³ = 0 + 1·x + 0·x² + (-1)·x³ + 0·x⁴ + 0·x⁵ + ...

So:

a₀ = 0

a₁ = 1

a₂ = 0

a₃ = -1

a₄ = 0, a₅ = 0, ... all higher terms = 0

0 is also a finite value.

Last verified on 04-06-2026

Explanation

Formula: Ā × B̄ = |Ā||B̄| sinθ 

For non-zero vectors, |Ā||B̄| is never 0.  

So cross product = 0 only when sinθ = 0 → θ = 0° or 180°

That’s exactly same direction or opposite direction = parallel.  

This works for ALL parallel cases.

So correct answer is = Ā × B̄ = 0̄

Last verified on 04-06-2026

Explanation

x + 2 is a factor = Put x = -2 in polynomial. If answer comes 0, then yes it’s a factor.

x⁴ + 2x³ - x - 2 → 16 - 16 + 2 - 2 = 0 → 0

Last verified on 04-06-2026

Explanation

f(x) = e^x - ax makes a U-shape because e^x is always curving up. 

U-shape has only 1 lowest point at x = ln a. 

So the minimum value = f(ln a) = a - a ln a

Last verified on 04-06-2026