The function $f$ is defined over the real numbers except for $x=0$. This table gives select values of $f$.
\begin{tabular}{lccccccc}
$x$ & -0.1 & -0.01 & -0.001 & 0 & 0.001 & 0.01 & 0.1 \
\hline$f(x)$ & 2.63 & 2.77 & 2.89 & undefined & 2.88 & 2.75 & 2.6
\end{tabular}

What is a reasonable estimate for $\lim _{x ightarrow 0} f(x)$ ?
Choose 1 answer:
(A) 0
(B) 0.1
(C) 2.75
(D) 2.9
(E) The limit doesn't exist

Question

The function $f$ is defined over the real numbers except for $x=0$. This table gives select values of $f$.
\begin{tabular}{lccccccc}
$x$ & -0.1 & -0.01 & -0.001 & 0 & 0.001 & 0.01 & 0.1 \
\hline$f(x)$ & 2.63 & 2.77 & 2.89 & undefined & 2.88 & 2.75 & 2.6
\end{tabular}

What is a reasonable estimate for $\lim _{x ightarrow 0} f(x)$ ?
Choose 1 answer:
(A) 0
(B) 0.1
(C) 2.75
(D) 2.9
(E) The limit doesn't exist

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We need to estimate the limit of $f(x)$ as $x$ approaches 0 using the given table of values. ‖ ‖ step 2 ⋮ The table provides the following values for $f(x)$: $$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline x & -0.1 & -0.01 & -0.001 & 0 & 0.001 & 0.01 & 0.1 \ \hline f(x) & 2.63 & 2.77 & 2.89 & \text{undefined} & 2.88 & 2.75 & 2.6 \ \hline \end{array} $$ ‖ ‖ step 3 ⋮ As $x$ approaches 0 from both the left and the right, the values of $f(x)$ are getting closer to a certain number. ‖ ‖ step 4 ⋮ From the left ($x = -0.001, -0.01, -0.1$), $f(x)$ values are 2.89, 2.77, and 2.63 respectively. From the right ($x = 0.001, 0.01, 0.1$), $f(x)$ values are 2.88, 2.75, and 2.6 respectively. ‖ ‖ step 5 ⋮ Both sides are approaching a value close to 2.75 as $x$ gets closer to 0. ‖ ‖ step 6 ⋮ Therefore, a reasonable estimate for $\lim_{x \to 0} f(x)$ is 2.75. ‖ ***C*** ∻Key Concept∻ ⚹Limit of a function⚹ ∻Explanation∻ ⚹The limit of a function as $x$ approaches a certain value is the value that the function's output gets closer to as $x$ gets closer to that value.⚹◊What is the definition of a limit of a function as x approaches a specific value?⍭ Generate me a similar question◊

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