91Ó°ÊÓ

Numerical methods involve using approximations with values close to the point of interest. Here, we started by evaluating \(\frac{2-\sqrt{x}}{4-x}\) for values of \(\boldsymbol{x \) near 4. This gives us an idea of the trend in the function's behavior and helps us estimate the limit.


Let's break it down step by step:
1. Start with a table from \(\boldsymbol{\Delta \)Tbl} = 0.1. Compute values for \(\boldsymbol{x} = 3.9, 3.95, 3.99, 4.01, 4.05, 4.1\).
2. Next, create a more precise table with \(\boldsymbol{\Delta \tbl} = 0.01\). Evaluate \(\boldsymbol{x} = 3.99, 3.995, 3.999, 4.001, 4.005, 4.01\).
3. Increase precision further with \(\boldsymbol{\Delta \tbl} = 0.001\). Calculate for \(\boldsymbol{x} = 3.999, 3.9995, 3.9999, 4.0001, 4.0005, 4.001\).
4. Achieve highest precision with \(\boldsymbol{\Delta \tbl} = 0.0001\). Compute for \(\boldsymbol{x} = 3.9999, 3.99995, 3.99999, 4.00001, 4.00005, 4.0001\).


Each of these steps brings us closer to understanding the function's behavior around \(\boldsymbol{x = 4}\). We observe that as \(\boldsymbol{x} \) gets closer to 4, the value of the function appears to approach \(\boldsymbol{-\frac{1}{4}}\).

Graphical methods involve visualizing the function by plotting it on a graph. This way, we can see the behavior of the function around the point where we want to find the limit.


Here's how to approach it:
1. Graph the function \(\boldsymbol{f(x) = \frac{2 - \sqrt{x}}{4 - x}}\). Use a graphing calculator or graphing software for precision.
2. Use the TRACE feature around \(\boldsymbol{x = 4}\) to closely examine the function.


You will notice the curve of the function. As \(\boldsymbol{x \) approaches 4 from both left and right sides, the value of the function \(\boldsymbol{f(x)}\) seems to approach \(\boldsymbol{-\frac{1}{4}}\). This visual verification is very useful as it confirms our numerical findings.


Graphical methods are essential for a quick, intuitive understanding of the function's behavior near specific points. They complement numerical and algebraic methods very well.

Algebraic techniques offer a precise way to find limits by manipulating the given expression. These methods aim to simplify the expression or handle indeterminate forms such as \(\boldsymbol{\frac{0}{0}}\).


Let’s go through the step-by-step algebraic verification of the limit:
1. Simplify the function: \(\boldsymbol{\frac{2 - \sqrt{x}}{4 - x}}\). We notice an indeterminate form as \(\boldsymbol{x \rightarrow 4} \), so we multiply the numerator and denominator by the conjugate of the numerator: \(\boldsymbol{2 + \sqrt{x}}\).
2. This leads to: \(\boldsymbol{\frac{(2-\sqrt{x})(2+\sqrt{x})}{(4-x)(2+\sqrt{x})} = \frac{4 - x}{(4-x)(2+\sqrt{x})}}\).
3. We simplify to: \(\boldsymbol{\frac{1}{-(2+\sqrt{x})}}\).
4. Now, evaluate the limit as \(\boldsymbol{x \rightarrow 4}\): this gives \(\boldsymbol{\frac{1}{-(2 + 2)} = -\frac{1}{4}}\).


These algebraic steps confirm our numerical and graphical findings. It is always reassuring to have multiple methods leading to the same result. By mastering algebraic techniques, we gain a deeper understanding of the inner workings of calculus.