Question

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

Accepted Answer

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 \u003e -1. This means the function has two different expressions depending on the value of x relative to -1. 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 \u003e -1, plot the horizontal line y = 3. Note the point of transition at x = -1. Step 3: Determine f(a). Since a = -1, use the expression for x ≤ -1, which is f(x) = $$x^2 + 1. $$Therefore, f(-1) = $$(-1)^2 + 1 = 2. $$Step 4: Evaluate the left-hand limit, $$ \lim_{x 	o -1^-} f(x) . As x $$approaches -1 from the left, f(x) follows the expression $$x^2 + 1. $$Thus, $$ \lim_{x 	o -1^-} f(x) = (-1)^2 + 1 = 2 . $$Step 5: Evaluate the right-hand limit, $$ \lim_{x 	o -1^+} f(x) . As x $$approaches -1 from the right, f(x) is constant at 3. Thus, $$ \lim_{x 	o -1^+} f(x) = 3 .$$

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 video answer for a similar problem:

Key Concepts

Piecewise Functions

Limits

Continuity

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≤−13 if x>−1; a=−1

Verified video answer for a similar problem:

Key Concepts

Piecewise Functions

Limits

Continuity

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