1, 2, 3 Again

Calculus Level 1

f ( x ) = ( x − 1 ) ( x − 3 ) ( x − 2 ) 2 f(x) = \dfrac{(x-1)(x-3)}{(x-2)^2}

Consider the function above, what is the value of lim ⁡ x → ∞ f ( x ) \displaystyle \lim_{x \to \infty} f(x) ?


The answer is 1.

Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

f ( x ) = ( x − 1 ) ( x − 3 ) ( x − 2 ) 2 = x 2 − 4 x + 3 x 2 − 4 x + 4 = 1 − 1 x 2 − 4 x + 4 f(x) = \dfrac{(x-1)(x-3)}{(x-2)^2} = \dfrac{x^2 - 4x +3}{x^2 - 4x +4} = 1 - \dfrac{1}{x^2 - 4x +4} .

As expanded above, when x x approaches infinity, the value of 1 x 2 − 4 x + 4 \dfrac{1}{x^2 - 4x +4} will reach 0 0 , and as shown in the graph below, the limit of this function is, therefore, 1 1 .

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Or you just divide numerator and denominator by the highest power in the denominator.

Hobart Pao - 5 years, 4 months ago

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Exactly. I just had a similar problem 1, 2, 3, Go! , so I thought I'd just reuse it for a new topic. ;)

Worranat Pakornrat - 5 years, 4 months ago

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