Question

Find the limit

lim x → 2 (x² - 4) / (x³ - 4x² +4x)
​

Answer 1

SOLUTION

TO DETERMINE

The value of

$\displaystyle \sf{\lim_{x \to 2} \: \frac{x^2-4}{ {x}^{3} - 4 {x}^{2} + 4x}}$

EVALUATION

$\displaystyle \sf{\lim_{x \to 2} \: \frac{x^2-4}{ {x}^{3} - 4 {x}^{2} + 4x}}$

$\displaystyle \sf{ = \lim_{x \to 2} \: \frac{x^2- {2}^{2} }{x( {x}^{2} - 4 x + 4)}}$

$\displaystyle \sf{ = \lim_{x \to 2} \: \frac{(x + 2)(x - 2) }{x {(x - 2)}^{2} }}$

$\displaystyle \sf{ = \lim_{x \to 2} \: \frac{(x + 2) }{x {(x - 2)}}}$

$= + \infty$

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Answer 2

Step-by-step explanation:

SOLUTION

TO DETERMINE

The value of

\displaystyle \sf{\lim_{x \to 2} \: \frac{x^2-4}{ {x}^{3} - 4 {x}^{2} + 4x}}

x→2

lim

x

3

−4x

2

+4x

x

2

−4

EVALUATION

\displaystyle \sf{\lim_{x \to 2} \: \frac{x^2-4}{ {x}^{3} - 4 {x}^{2} + 4x}}

x→2

lim

x

3

−4x

2

+4x

x

2

−4

\displaystyle \sf{ = \lim_{x \to 2} \: \frac{x^2- {2}^{2} }{x( {x}^{2} - 4 x + 4)}}=

x→2

lim

x(x

2

−4x+4)

x

2

−2

2

\displaystyle \sf{ = \lim_{x \to 2} \: \frac{(x + 2)(x - 2) }{x {(x - 2)}^{2} }}=

x→2

lim

x(x−2)

2

(x+2)(x−2)

\displaystyle \sf{ = \lim_{x \to 2} \: \frac{(x + 2) }{x {(x - 2)}}}=

x→2

lim

x(x−2)

(x+2)

= + \infty=+∞

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