Question

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

Short answer

Answer: The values of the limits are $\underset{x\to 4}{lim}f(x)=e^2$ and $\underset{x\to 0^{+}}{lim}f(x)=1$.

Step-by-step solution

Determine the interval of continuity for $f(x)=e^{\sqrt{x}}$

To determine the interval of continuity, we need to first find the domain of the function. Since $f(x)=e^{\sqrt{x}}$, the only restriction on the domain is that the radicand of the square root, $x$, must be greater than or equal to 0. Therefore, the domain of the function is $x\ge 0$. Now, we know that the exponential function is continuous for all real numbers. Since there are no restrictions on the domain other than the non-negativity of $x$, the function is continuous for all $x\ge 0$. Thus, the interval of continuity is $[0,\mathrm{\infty })$.

Analyze $\underset{x\to 4}{lim}f(x)$

Since $f(x)$ is continuous on the interval $[0,\mathrm{\infty })$, we can evaluate the limit $\underset{x\to 4}{lim}f(x)$ by plugging in $x=4$ into the function: $$f(4)=e^{\sqrt{4}}=e^2$$ Thus, the limit is: $$\underset{x\to 4}{lim}f(x)=e^2$$

Analyze $\underset{x\to 0^{+}}{lim}f(x)$

Similar to Step 2, the function is continuous on its domain $[0,\mathrm{\infty })$, so we can evaluate the limit $\underset{x\to 0^{+}}{lim}f(x)$ by plugging in $x=0$: $$f(0)=e^{\sqrt{0}}=e^0=1$$ Thus, the limit is: $$\underset{x\to 0^{+}}{lim}f(x)=1$$ To sum up, the function $f(x)=e^{\sqrt{x}}$ is continuous on the interval $[0,\mathrm{\infty })$. The limits are evaluated as follows: - $\underset{x\to 4}{lim}f(x)=e^2$ - $\underset{x\to 0^{+}}{lim}f(x)=1$

Key concepts

Limits of Functions

The concept of a limit is foundational in calculus, serving as the cornerstone for understanding continuous functions and analyzing function behavior around specific points. When we talk about the limit of a function as the input approaches a certain value, we're describing what the output of the function is converging to, without necessarily reaching that point.

For the function in our exercise, $f(x)=e^{\sqrt{x}}$, evaluating limits like $\underset{x\to 4}{lim}f(x)$ and $\underset{x\to 0^{+}}{lim}f(x)$ is straightforward due to the function's continuity – which will be discussed in the following section – and allows us to simply substitute the value of $x$ into $f(x)$ to find the limits directly.

Continuous Functions

A continuous function can be visualized as a graph that can be drawn without lifting the pencil from the paper. Formally, this means that for every point within the function's domain, the limit of the function as you approach that point equals the actual value of the function at that point.

With our example function, $f(x)=e^{\sqrt{x}}$, it remains uninterrupted as $x$ increases from 0 to infinity, exhibiting a seamless behavior characteristic of continuous functions. Therefore, it's described as continuous on the interval $[0,\mathrm{\infty })$, and the limits can be assessed by directly substituting the values of $x$.

Exponential Functions

Exponential functions, like our $f(x)=e^{\sqrt{x}}$, have the form $f(x)=a^{g(x)}$, where $a$ is a positive constant, and $g(x)$ is an expression involving $x$. These functions are known for their rapid growth or decay characteristics.

The base $e$ stands out since it is the natural exponential base, commonly encountered in growth and decay problems, and has special properties that make calculus work well. In our case, the continuously compounding quality of $e$ ensures that the function grows smoothly and without any interruptions.

Function Domain Analysis

Analyzing the domain of a function involves identifying the set of all possible input values for which the function is defined. This is essential for determining a function’s continuity over its domain. For the function $f(x)=e^{\sqrt{x}}$, the square root demands the input $x$ to be non-negative, thus restricting the domain to $x\ge 0$. Because of this, the domain analysis concludes that the function is continuous on its natural domain $[0,\mathrm{\infty })$.

In essence, a thorough function domain analysis is vital for understanding where to expect continuous behavior and consequently, where a function can be assessed using limits and continuous function properties.