Sketch a graph of f and use it to make a conjecture about the values of f(a), lim x→a^−f(x),lim x→a^+f(x), and lim x→a f(x) or state that they do not exist.
f(x) = {√x if x<4
3 if x=4; a=4
x+1 if x>4

Verified step by step guidance

Step 1: Understand the piecewise function f(x). It is defined as f(x) = $\sqrt{x}$ for x < 4, f(x) = 3 for x = 4, and f(x) = x + 1 for x > 4.

Step 2: Sketch the graph of each piece of the function. For x < 4, plot the curve of $\sqrt{x}$, which is a part of the square root function starting from the origin and increasing as x approaches 4.

Step 3: At x = 4, plot the point (4, 3) since f(x) = 3 when x = 4. This is a single point on the graph.

Step 4: For x > 4, plot the line x + 1, which starts at (4, 5) and increases linearly. Note that this line does not include the point (4, 5) since x > 4.

Step 5: Analyze the graph to make conjectures about the limits. Observe the behavior of the function as x approaches 4 from the left and right, and compare these to the value of f(4).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, f(x) has three distinct definitions depending on whether x is less than, equal to, or greater than 4. Understanding how to evaluate and graph these segments is crucial for analyzing the function's behavior at specific points.

Limits

Limits describe the behavior of a function as the input approaches a certain value. The left-hand limit (lim x→a^−f(x)) and right-hand limit (lim x→a^+f(x)) are essential for determining continuity at a point. If both limits exist and are equal, the overall limit (lim x→a f(x)) exists and equals that common value.

Continuity

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. For f(x) at x=4, we need to check if lim x→4^−f(x) equals lim x→4^+f(x) and if both equal f(4). If they do not match, the function is discontinuous at that point, which affects the conjecture about the function's values.

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Textbook Question

Determine $$\lim$_{x$\rightarrow\]\infty$}f$\left$(x$\right$)$ and $$\lim$_{x$\rightarrow$-$\infty$}f$\left$(x$\right$)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).