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The limit of a function is a fundamental concept in calculus. It describes the behavior of a function as its input approaches a particular value. When dealing with derivatives, limits help us understand how a function changes at precise points.
To find the derivative of a function using its definition, we use the formula:

• \( f'(t) = \lim_{h \to 0} \frac{f(t+h) - f(t)}{h} \)
  This formula represents the slope of the tangent line to the function at any point \( t \).

In our exercise, the limit finds how the function \( g(t) = \frac{1}{\sqrt{t}} \) changes as \( t \) becomes infinitesimally small. The limit process captures the instantaneous rate of change—or the derivative—of the function.

Simplifying expressions is crucial for evaluating limits, especially when the initial form is complex or indeterminate.
Let's consider simplifying the expression \( \frac{1}{\sqrt{t+h}} - \frac{1}{\sqrt{t}} \). This step involves

• First finding a common denominator, resulting in \( \frac{\sqrt{t} - \sqrt{t+h}}{\sqrt{t(t+h)}} \).
• Then, multiply the numerator and denominator by the conjugate \( \sqrt{t} + \sqrt{t+h} \). This trick helps eliminate radicals which simplifies the work in limits.

By doing these steps, the expression simplifies to a form where cancelling terms is possible, allowing us to evaluate the limit more easily.

Function domain is about finding all the possible input values (or x-values) for which a function is defined. For certain functions, the domain might be restricted due to operations like division by zero or taking the square root of a negative number.
In our function \( g(t) = \frac{1}{\sqrt{t}} \), the square root limits the domain to values greater than zero \( t > 0 \) as the square root of a negative number is not defined in the real numbers.
Similarly, the domain of its derivative \( g'(t) = \frac{-1}{2t^{3/2}} \) also requires \( t > 0 \). This is because the term \( t^{3/2} \) in the denominator must remain positive to avoid division by zero or trying to take a square root of a negative.

Differentiation techniques present various methods to find the derivative of a function. While using the definition of a derivative is foundational, other techniques often streamline the process.
In this particular exercise, we focus on utilizing the limit definition. This approach involves:

• Setting up and simplifying a difference quotient after substituting function values
• Applying algebraic tricks like using conjugates to simplify expressions further
• Evaluating the resulting limit as \( h \to 0 \)

These steps help transform complex expressions into more manageable forms, allowing for a clear calculation of the derivative. Advanced techniques like the power rule, chain rule, or product rule can make the differentiation process quicker in other contexts.