They Are Just Two Different Functions

Calculus Level 2

True or false :

lim ⁡ x → 0 x 2 x = lim ⁡ x → 0 x 2 lim ⁡ x → 0 x . \large \lim_{x\to0} \dfrac{x^2}x = \dfrac{\displaystyle \lim_{x\to0} x^2 }{\displaystyle \lim_{x\to0} x} \; .

True False

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Pi Han Goh by Pi Han Goh , 30, Malaysia

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1 solution

lim ⁡ x → 0 x 2 x = lim ⁡ x → 0 x = 0 \displaystyle \lim_{x \to 0}{\dfrac{x^2}{x}} = \displaystyle \lim_{x \to 0}{x}=0 .

lim ⁡ x → 0 x 2 lim ⁡ x → 0 x = 0 0 \dfrac{\displaystyle \lim_{x \to 0}{x^2}}{\displaystyle \lim_{x \to 0}{x}} = \dfrac{0}{0} which is undefined.

Hence the answer is false.

7 Helpful 0 Interesting 0 Brilliant 0 Confused

Exactly. We need to evaluate both the limits in the fraction first. But since they are both equal to 0, then their ratio, 0/0 is indeterminate, so RHS is indeterminate, whereas LHS is simply equal to 0.

Is the following equation false as well?

lim ⁡ x → 0 x 2 x = lim ⁡ y → 0 lim ⁡ x → y x 2 lim ⁡ x → y x \large \lim_{x\to0} \dfrac{x^2}x =\lim_{y\to0} \dfrac{\displaystyle \lim_{x\to y} x^2 }{\displaystyle \lim_{x\to y} x}

Pi Han Goh - 5 years, 2 months ago

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I think that it is true.

L.H.S is 0.

R.H.S = lim ⁡ y → 0 y 2 y = 0 = \displaystyle \lim_{y \to 0}{\dfrac {y^2} y }= 0 .

A Former Brilliant Member - 5 years, 2 months ago

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