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Graphical Display Calculators (GDCs) are powerful tools that allow students to estimate limits by visualizing how functions behave as certain variables approach specific values. When using a GDC to approximate limits, you can directly input expressions and watch how their values change for sequences converging to a point of interest.

In this case, you're estimating the limit \( \lim_{h \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h} \). To do this with a GDC:

• Set "\(x\)" to a real number, e.g., 1.
• Substitute progressively smaller values for \(h\), like 0.1, 0.01, 0.001.
• Observe the values output for the expression \(\frac{\sqrt{x+h}-\sqrt{x}}{h}\).

As \(h\) gets smaller, the results on your GDC should approach \(\frac{1}{2\sqrt{x}}\). This tool helps you visualize and understand how the expression behaves, making the concept of limits more intuitive.

Analytic methods for evaluating limits go beyond approximations to calculate exact values. By manipulating algebraic expressions systematically, you can rigorously prove limit results. The goal is to handle the algebraic form directly to simplify or transform it until a limit can be assessed.

For the problem at hand, start with:\( \frac{\sqrt{x+h}-\sqrt{x}}{h} \)Analytically, we need to clear out the indeterminate form ("0/0") as \(h\) approaches zero. Such manipulation requires deeper understanding of algebra and calculus operations like rationalization and simplification.

Rationalizing is a crucial step in eliminating indeterminacies when evaluating limits involving radicals. By multiplying by a form of 1, often using the conjugate, we can simplify expressions to solve limits analytically.

In this scenario, you rationalize the numerator of\( \frac{\sqrt{x+h}-\sqrt{x}}{h} \)by multiplying both the numerator and the denominator by the conjugate:\( \sqrt{x+h}+\sqrt{x} \).This leads to:\[ \frac{(x+h) - x}{h(\sqrt{x+h}+\sqrt{x})} = \frac{h}{h(\sqrt{x+h}+\sqrt{x})} \]Where the \(h\) terms cancel each other out, simplifying to\( \frac{1}{\sqrt{x+h} + \sqrt{x}} \).

By allowing \(h\) to approach 0, this expression simplifies to the desired limit, showcasing how rationalization clears up the original indeterminate form and helps in finding the exact solution \( \frac{1}{2\sqrt{x}} \). This process is key to mastering calculus methods for solving limits.