Find lim_(x rarr4)(x^(3)-4x^(2))/(x^(2)-16). Choose 1 answer: (A) 2 (B) 1 (c) -4 (D) The limit doesn't exist

Identify Function and Point: Identify the function and the point at which we need to find the limit. The function is (x^3 - 4x^2) / (x^2 - 16) and we need to find the limit as x approaches 4.Attempt Direct Substitution: Attempt to directly substitute x = 4 into the function to see if the limit can be evaluated this way. Substitute x = 4 into (x^3 - 4x^2) / (x^2 - 16) to check if the function is defined at this point. ((4)^3 - 4(4)^2) / ((4)^2 - 16) = (64 - 64) / (16 - 16) = 0 / 0 We get an indeterminate form 0 / 0, which means we cannot directly evaluate the limit by substitution.Factor and Simplify Expression: Factor the numerator and denominator to simplify the expression and potentially cancel out common factors. The numerator x^3 - 4x^2 can be factored as x^2(x - 4). The denominator x^2 - 16 can be factored as (x + 4)(x - 4). Now the function is (x^2(x - 4)) / ((x + 4)(x - 4)).Cancel Common Factor: Cancel out the common factor (x - 4) from the numerator and denominator. After canceling, the function simplifies to x^2 / (x + 4).Evaluate Limit: Now that the function is simplified, try to directly substitute x = 4 again. Substitute x = 4 into x^2 / (x + 4) to evaluate the limit. (4)^2 / (4 + 4) = 16 / 8 = 2 The limit as x approaches 4 is 2.

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Find
lim_(x rarr4)(x^(3)-4x^(2))/(x^(2)-16).
Choose 1 answer:
(A) 2
(B) 1
(c) -4
(D) The limit doesn't exist

Find $\lim _{x \rightarrow 4} \frac{x^{3}-4 x^{2}}{x^{2}-16}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $2$ $\newline$ (B) $1$ $\newline$ (C) $-4$ $\newline$ (D) The limit doesn't exist

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Math Problems

Algebra 2

Evaluate exponential functions

Full solution

Q. Find

\lim _{x \rightarrow 4} \frac{x^{3}-4 x^{2}}{x^{2}-16}

\newline

Choose

1

answer:

\newline

(A)

2

\newline

(B)

1

\newline

(C)

-4

\newline

(D) The limit doesn't exist

Identify Function and Point: Identify the function and the point at which we need to find the limit. The function is $(x^3 - 4x^2) / (x^2 - 16)$ and we need to find the limit as $x$ approaches $4$ .
Attempt Direct Substitution: Attempt to directly substitute $x = 4$ into the function to see if the limit can be evaluated this way. $\newline$ Substitute $x = 4$ into $(x^3 - 4x^2) / (x^2 - 16)$ to check if the function is defined at this point. $\newline$ $((4)^3 - 4(4)^2) / ((4)^2 - 16) = (64 - 64) / (16 - 16) = 0 / 0$ $\newline$ We get an indeterminate form $0 / 0$ , which means we cannot directly evaluate the limit by substitution.
Factor and Simplify Expression: Factor the numerator and denominator to simplify the expression and potentially cancel out common factors. $\newline$ The numerator $x^3 - 4x^2$ can be factored as $x^2(x - 4)$ . $\newline$ The denominator $x^2 - 16$ can be factored as $(x + 4)(x - 4)$ . $\newline$ Now the function is $(x^2(x - 4)) / ((x + 4)(x - 4))$ .
Cancel Common Factor: Cancel out the common factor $(x - 4)$ from the numerator and denominator. $\newline$ After canceling, the function simplifies to $\frac{x^2}{(x + 4)}$ .
Evaluate Limit: Now that the function is simplified, try to directly substitute $x = 4$ again. $\newline$ Substitute $x = 4$ into $\frac{x^2}{x + 4}$ to evaluate the limit. $\newline$ $\frac{(4)^2}{4 + 4} = \frac{16}{8} = 2$ $\newline$ The limit as $x$ approaches $4$ is $2$ .