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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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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. $\newline$ $\lim_{x \to 2}(x - 2)(x + 2)^2 = (2 - 2)(2 + 2)^2$ $\newline$ $= 0 \times 4^2$ $\newline$ $= 0$

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\newline

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f(x)

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\newline

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\newline

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\newline

Simplify any fractions.

\newline

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Question

p(x)

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are differentiable.

\newline

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u(x)

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Question

a(x)

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\newline

c(x)

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\newline

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\newline

Simplify any fractions.

\newline

c'(2)=

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Question

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\newline

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\newline

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o'(7)

\newline

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\newline

o'(7)=

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