Complete the table by computing f(x) at the given values of...

Complete the table by computing $f(x)$ at the given values of $x .$ Use these results to estimate the indicated limit (if it exists). $$f(x)=x^{2}+1 ; \lim _{x \rightarrow 2} f(x)$$ $$\begin{array}{lllllll}\hline x & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & & & & & \\\hline\end{array}$$

Complete the table by computing $f(x)$ at the given values of $x .$ Use these results to estimate the indicated limit (if it exists).
$$f(x)=x^{2}+1 ; \lim _{x \rightarrow 2} f(x)$$ $$\begin{array}{lllllll}\hline x & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \\
\hline f(x) & & & & & & \\\hline\end{array}$$

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Key Concepts

Numerical Estimation via Tables

Numerical estimation through tables involves calculating function values at inputs that approach the target point from both sides. This technique provides an empirical way to observe the trend of function values, helping to confirm the limit and understand the function's behavior near that point.

Direct Substitution

Direct substitution is a method used to evaluate limits when the function is continuous at the point of interest. By substituting the target value into the function, one can immediately obtain the value of the limit, demonstrating the seamless behavior of continuous functions.

Limit

The limit of a function at a point describes the value that the function approaches as the input gets arbitrarily close to that point, regardless of the actual function value at the point. This concept is fundamental in calculus and is used to analyze the behavior of functions near specific inputs.

Continuity

Continuity is a property of a function where small changes in the input lead to small changes in the output, without sudden jumps. For continuous functions, the limit as x approaches a particular value is equal to the function’s actual value at that point, which simplifies the evaluation of limits.

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