Question

Determine the interval(s) on which the following functions are continuous; then evaluate the given limits. $$f(x)=e^{\sqrt{x}} ; \lim _{x \rightarrow 4} f(x) ; \lim _{x \rightarrow 0^{+}} f(x)$$

Short answer

Question: Determine the intervals of continuity for the function \(f(x) = e^{\sqrt{x}}\) and find the limits \(\lim_{x \rightarrow 4} f(x)\) and \(\lim_{x \rightarrow 0^{+}} f(x)\). Answer: The function \(f(x) = e^{\sqrt{x}}\) is continuous on the interval \([0, +\infty)\). The limit \(\lim _{x \rightarrow 4} f(x) = e^2\), and the limit \(\lim _{x \rightarrow 0^{+}} f(x) = 1\).

Step-by-step solution

Determine the domain of the function

The function is given by \(f(x)=e^{\sqrt{x}}\). The square root inside the exponent is only defined for non-negative values of x, so the domain of the function is \([0, +\infty)\).

Determine the continuity of the function

The exponential function (\(e^x\)) is continuous for all real numbers, and the square root function (\(\sqrt{x}\)) is continuous for non-negative values of x. As a result, we can conclude that the composition of these two functions, \(f(x)=e^{\sqrt{x}}\), is continuous on the domain \([0, +\infty)\).

Evaluate the limit as x approaches 4

Since the function is continuous on its entire domain, we can evaluate the limit by directly substituting the value of x = 4 into the function: $$\lim _{x \rightarrow 4} f(x) = \lim_{x \rightarrow 4} e^{\sqrt{x}} = e^{\sqrt{4}}=e^2$$

Evaluate the limit as x approaches 0 from the right

Similarly, we can evaluate this limit by directly substituting the value of x = 0 into the function: $$\lim _{x \rightarrow 0^{+}} f(x) = \lim_{x \rightarrow 0} e^{\sqrt{x}} = e^{\sqrt{0}}=e^0=1$$ In conclusion, the function \(f(x) = e^{\sqrt{x}}\) is continuous on the interval \([0, +\infty)\). The limit \(\lim _{x \rightarrow 4} f(x) = e^2\), and the limit \(\lim _{x \rightarrow 0^{+}} f(x) = 1\).

Key concepts

Limits

In mathematics, limits help us understand the behavior of functions as they approach a certain point. The limit of a function describes where the function's value is heading as the input reaches a particular point or approaches infinity. For instance, the statement \( \lim_{x \rightarrow a} f(x) = L \) means that as \( x \) gets closer and closer to \( a \), \( f(x) \) gets closer to \( L \). Limits are crucial for understanding concepts such as continuity and differentiability of functions. Evaluating limits, particularly at points where functions are continuous, can often be done by direct substitution, as seen in this exercise. For example, we found that \( \lim_{x \rightarrow 4} e^{\sqrt{x}} = e^{2} \) simply by substituting \( x = 4 \) into the function. Sometimes calculating a limit is not straightforward; this is especially true at points of discontinuity or when a function tends towards infinity or zero from one side. Therefore, it's important to understand how to approach these calculations using techniques like factorization, rationalization, or L'Hôpital's Rule for more complex cases.

Domain of a Function

The domain of a function is the set of all possible input values \( x \) for which the function is defined. For any function, it's crucial to identify the domain to explore where the function can operate without issues. In this particular exercise, the function \( f(x) = e^{\sqrt{x}} \) depends on the square root of \( x \), which is only defined for non-negative values. As a result, the domain of this function is the interval \([0, +\infty)\). This means any value from 0 onwards can be input into the function without making it undefined. Understanding the domain is essential not only for evaluating limits but also for understanding the continuity and behavior of the function across its domain.To find the domain:

• Identify any expressions that might become undefined (e.g., division by zero, negative square roots).
• Ensure the validity of trigonometric, logarithmic, or exponential relationships.

This is an important first step in any problem of calculus or algebra involving functions.

Exponential Functions

Exponential functions are a broad class of functions characterized by constant growth rates. In a typical exponential function of the form \( f(x) = a^x \), the base \( a \) is a constant, and the variable \( x \) is the exponent. The natural exponential function, with the base \( e \approx 2.718 \), is particularly prevalent in mathematics due to its unique properties and natural occurrences.In this exercise, the function \( f(x) = e^{\sqrt{x}} \) is a composite function that involves an exponential function with the square root function as the exponent. Exponential functions are special because they:

• Are continuous and differentiable for all real numbers in their domains.
• Exhibit constant relative growth rates.
• Have unique characteristics making them suitable for modeling growth processes in nature, finance, and science.

They stretch along the y-axis very rapidly, differing from linear or polynomial functions. Understanding their behavior helps predict outputs based on various inputs, making them invaluable in many real-world applications.