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^2+1 if x≤−1
3 if x>−1; a=−1

Verified step by step guidance

1. Step 1: Identify the piecewise function. The function f(x) is defined as f(x) = x^2 + 1 for x ≤ -1 and f(x) = 3 for x > -1. This means the function has two different expressions depending on the value of x relative to -1.
2. Step 2: Sketch the graph of f(x). For x ≤ -1, plot the graph of y = x^2 + 1, which is a parabola opening upwards with its vertex at (0, 1). For x > -1, plot the horizontal line y = 3. Note the point of transition at x = -1.
3. Step 3: Determine f(a). Since a = -1, use the expression for x ≤ -1, which is f(x) = x^2 + 1. Therefore, f(-1) = (-1)^2 + 1 = 2.
4. Step 4: Evaluate the left-hand limit, $\underset{x\to -1^{-}}{lim}f(x)$. As x approaches -1 from the left, f(x) follows the expression x^2 + 1. Thus, $\underset{x\to -1^{-}}{lim}f(x)=(-1{)}^2+1=2$.
5. Step 5: Evaluate the right-hand limit, $\underset{x\to -1^{+}}{lim}f(x)$. As x approaches -1 from the right, f(x) is constant at 3. Thus, $\underset{x\to -1^{+}}{lim}f(x)=3$.

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, f(x) has two distinct definitions: one for x ≤ -1 and another for x > -1. Understanding how to evaluate piecewise functions at specific points is crucial for analyzing their behavior and limits.

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 and the overall limit (lim x→a f(x)). Evaluating these limits helps in understanding how the function behaves around the point of interest.

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 the function f(x) at a = -1, checking continuity involves comparing f(-1) with the left-hand and right-hand limits. If these values do not match, the function is discontinuous at that point.