Pregunta
Evaluate the following limits. \[ \lim _{x \rightarrow 2}\left(x^{3}-3 x^{2}+6 x-7\right) \] 1.
Ask by Marshall Marshall. in the United States
Feb 03,2025
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The limit is 1.
Solución
To evaluate the limit:
\[
\lim _{x \rightarrow 2}\left(x^{3}-3 x^{2}+6 x-7\right)
\]
Since the function \( f(x) = x^{3} - 3x^{2} + 6x - 7 \) is a **polynomial**, it is **continuous** everywhere. This means we can directly substitute \( x = 2 \) into the function to find the limit.
\[
f(2) = 2^{3} - 3(2)^{2} + 6(2) - 7
\]
Calculating each term:
\[
2^{3} = 8
\]
\[
3(2)^{2} = 3 \times 4 = 12
\]
\[
6(2) = 12
\]
Now, substitute these values back into the equation:
\[
f(2) = 8 - 12 + 12 - 7
\]
Simplify step by step:
\[
8 - 12 = -4
\]
\[
-4 + 12 = 8
\]
\[
8 - 7 = 1
\]
Therefore, the limit is:
\[
\boxed{1}
\]
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Simplifique esta solución The Deep Dive
To evaluate the limit \[ \lim _{x \rightarrow 2}\left(x^{3}-3 x^{2}+6 x-7\right), \] we can start by substituting \(x = 2\) directly into the polynomial expression. Calculating, \[ (2)^3 - 3(2)^2 + 6(2) - 7. \] First, we compute each term: - \(2^3 = 8\), - \(3(2^2) = 3(4) = 12\), - \(6(2) = 12\), - and then we subtract 7. Putting it all together: \[ 8 - 12 + 12 - 7 = 8 - 7 = 1. \] Thus, we find that \[ \lim _{x \rightarrow 2}\left(x^{3}-3 x^{2}+6 x-7\right) = 1. \] Therefore, the limit is \(\boxed{1}\).
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