compute the values of $f(x)=\frac{x-7}{(x-1)^2}$ in the table to the right and use them to determine $\lim_{x \to 1} f(x)$.

complete the table below.

$\begin{array}{|c|c|c|c|}
\hline
x & f(x) & x & f(x) \\
\hline
1.1 & & 0.9 & \\
\hline
1.01 & & 0.99 & \\
\hline
1.001 & & 0.999 & \\
\hline
1.0001 & & 0.9999 & \\
\hline
\end{array}$

(type integers or decimals.)

Question

compute the values of $f(x)=\frac{x-7}{(x-1)^2}$ in the table to the right and use them to determine $\lim_{x \to 1} f(x)$.

complete the table below.

$\begin{array}{|c|c|c|c|}
\hline
x & f(x) & x & f(x) \\
\hline
1.1 & & 0.9 & \\
\hline
1.01 & & 0.99 & \\
\hline
1.001 & & 0.999 & \\
\hline
1.0001 & & 0.9999 & \\
\hline
\end{array}$

(type integers or decimals.)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Numerical Limit Estimation", "Infinite Limits" ], "current_concepts": [ "Numerical Limit Estimation", "Infinite Limits", "Rational Functions" ] } ## Evaluate function values for $x > 1$ $$ \begin{aligned} f(1.1) &= \frac{1.1 - 7}{(1.1 - 1)^2} = \frac{-5.9}{0.01} = -590 \ f(1.01) &= \frac{1.01 - 7}{(1.01 - 1)^2} = \frac{-5.99}{0.0001} = -59900 \ f(1.001) &= \frac{1.001 - 7}{(1.001 - 1)^2} = \frac{-5.999}{0.000001} = -5999000 \ f(1.0001) &= \frac{1.0001 - 7}{(1.0001 - 1)^2} = \frac{-5.9999}{0.00000001} = -599990000 \end{aligned} $$ ## Evaluate function values for $x < 1$ $$ \begin{aligned} f(0.9) &= \frac{0.9 - 7}{(0.9 - 1)^2} = \frac{-6.1}{0.01} = -610 \ f(0.99) &= \frac{0.99 - 7}{(0.99 - 1)^2} = \frac{-6.01}{0.0001} = -60100 \ f(0.999) &= \frac{0.999 - 7}{(0.999 - 1)^2} = \frac{-6.001}{0.000001} = -6001000 \ f(0.9999) &= \frac{0.9999 - 7}{(0.9999 - 1)^2} = \frac{-6.0001}{0.00000001} = -600010000 \end{aligned} $$ ## Determine the limit as $x \to 1$ $$ \lim_{x \to 1} f(x) = -\infty $$ Complete the table below. | $x$ | $f(x)$ | $x$ | $f(x)$ | | :--- | :--- | :--- | :--- | | $1.1$ | -590 | $0.9$ | -610 | | $1.01$ | -59900 | $0.99$ | -60100 | | $1.001$ | -5999000 | $0.999$ | -6001000 | | $1.0001$ | -599990000 | $0.9999$ | -600010000 | The limit is: $$ \lim_{x \to 1} f(x) = -\infty $$ { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Numerical Limit Estimation" ] }

\(x\)	\(f(x)\)	\(x\)	\(f(x)\)
\(1.01\)	<blank>-59900</blank>	\(0.99\)	<blank>-60100</blank>
\(1.001\)	<blank>-5999000</blank>	\(0.999\)	<blank>-6001000</blank>
\(1.0001\)	<blank>-599990000</blank>	\(0.9999\)	<blank>-600010000</blank>

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QUESTION IMAGE

Question

Explanation:

Evaluate function values for \(x > 1\)

Evaluate function values for \(x < 1\)

Determine the limit as \(x \to 1\)

Answer: