Find lim_(x rarr2)(x^(4)-4x^(3)+4x^(2))/(x-2). Choose 1 answer: (A) -4 (B) 0 (c) 4 (D) The limit doesn't exist

Identify the form: Identify the form of the limit. We need to find the limit of the function as x approaches 2. Let's first substitute x = 2 into the function to see if we can directly evaluate the limit. lim_(x -> 2)(x^4 - 4x^3 + 4x^2) / (x - 2) = (2^4 - 4*2^3 + 4*2^2) / (2 - 2) = (16 - 32 + 16) / 0 = 0 / 0 This is an indeterminate form, so we cannot directly evaluate the limit.Factor the numerator: Factor the numerator. Since we have an indeterminate form, we should try to simplify the expression. The numerator is a polynomial that can be factored, especially since we see a common pattern (x^2 - 2x)^2. Let's factor the numerator: x^4 - 4x^3 + 4x^2 = (x^2)^2 - 2*2*x*x^2 + (2x)^2 = (x^2 - 2x)^2 Now we have: lim_(x -> 2)(x^2 - 2x)^2 / (x - 2)Factor out a common term: Factor out a common term. We notice that the numerator is a perfect square and can be written as (x - 2)^2 * (x + 2)^2. Since we have an (x - 2) term in the denominator, we can cancel out one (x - 2) term from the numerator and the denominator. lim_(x -> 2)(x - 2)^2 * (x + 2)^2 / (x - 2) = lim_(x -> 2)(x - 2)(x + 2)^2Evaluate the limit: Evaluate the limit. Now that we have canceled the common term, we can directly substitute x = 2 into the remaining expression to find the limit. lim_(x -> 2)(x - 2)(x + 2)^2 = (2 - 2)(2 + 2)^2 = 0 * 4^2 = 0

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Find
lim_(x rarr2)(x^(4)-4x^(3)+4x^(2))/(x-2).
Choose 1 answer:
(A) -4
(B) 0
(c) 4
(D) The limit doesn't exist

Find $\lim _{x \rightarrow 2} \frac{x^{4}-4 x^{3}+4 x^{2}}{x-2}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-4$ $\newline$ (B) $0$ $\newline$ (C) $4$ $\newline$ (D) The limit doesn't exist

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Algebra 1

Compare linear and exponential growth

Full solution

Q. Find

\lim _{x \rightarrow 2} \frac{x^{4}-4 x^{3}+4 x^{2}}{x-2}

\newline

Choose

1

answer:

\newline

(A)

-4

\newline

(B)

0

\newline

(C)

4

\newline

(D) The limit doesn't exist

Identify the form: Identify the form of the limit. $\newline$ We need to find the limit of the function as $x$ approaches $2$ . Let's first substitute $x = 2$ into the function to see if we can directly evaluate the limit. $\newline$ $\lim_{x \to 2}\frac{x^4 - 4x^3 + 4x^2}{x - 2} = \frac{2^4 - 4\cdot2^3 + 4\cdot2^2}{2 - 2}$ $\newline$ $= \frac{16 - 32 + 16}{0}$ $\newline$ $= \frac{0}{0}$ $\newline$ This is an indeterminate form, so we cannot directly evaluate the limit.
Factor the numerator: Factor the numerator. $\newline$ Since we have an indeterminate form, we should try to simplify the expression. The numerator is a polynomial that can be factored, especially since we see a common pattern $(x^2 - 2x)^2$ . $\newline$ Let's factor the numerator: $\newline$ $x^4 - 4x^3 + 4x^2 = (x^2)^2 - 2\cdot2\cdot x\cdot x^2 + (2x)^2$ $\newline$ $= (x^2 - 2x)^2$ $\newline$ Now we have: $\newline$ $\lim_{x \to 2}\frac{(x^2 - 2x)^2}{(x - 2)}$
Factor out a common term: Factor out a common term. $\newline$ We notice that the numerator is a perfect square and can be written as $(x - 2)^2 * (x + 2)^2$ . Since we have an $(x - 2)$ term in the denominator, we can cancel out one $(x - 2)$ term from the numerator and the denominator. $\newline$ $\lim_{x \to 2}(x - 2)^2 * (x + 2)^2 / (x - 2)$ $\newline$ = $\lim_{x \to 2}(x - 2)(x + 2)^2$
Evaluate the limit: Evaluate the limit. $\newline$ Now that we have canceled the common term, we can directly substitute $x = 2$ into the remaining expression to find the limit. $\newline$ $\lim_{x \to 2}(x - 2)(x + 2)^2 = (2 - 2)(2 + 2)^2$ $\newline$ $= 0 \times 4^2$ $\newline$ $= 0$