Let​​ f(x) = {x^2+1 / if x<−1
√x+1 if x≥−1.


Compute the following limits or state that they do not exist.
limx→−1 f(x)

Verified step by step guidance

1. <insert step 1> Identify the piecewise function f(x) and the point of interest, which is x = -1. The function is defined as f(x) = x^2 + 1 for x < -1 and f(x) = \sqrt{x} + 1 for x \geq -1.
2. <insert step 2> To find the limit as x approaches -1, consider the left-hand limit (as x approaches -1 from the left) and the right-hand limit (as x approaches -1 from the right) separately.
3. <insert step 3> Calculate the left-hand limit: lim_{x \to -1^-} f(x) = lim_{x \to -1^-} (x^2 + 1). Substitute x = -1 into the expression x^2 + 1 to find the left-hand limit.
4. <insert step 4> Calculate the right-hand limit: lim_{x \to -1^+} f(x) = lim_{x \to -1^+} (\sqrt{x} + 1). Substitute x = -1 into the expression \sqrt{x} + 1 to find the right-hand limit.
5. <insert step 5> Compare the left-hand and right-hand limits. If they are equal, the limit exists and is equal to this common value. If they are not equal, the limit does not exist.

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 formulas: one for x < -1 and another for x ≥ -1. Understanding how to evaluate piecewise functions is crucial for determining limits at points where the function's definition changes.

Limits

A limit describes the behavior of a function as the input approaches a certain value. To compute the limit of f(x) as x approaches -1, we need to evaluate the function from both sides of -1, using the appropriate piece of the function for each side. This helps us determine if the limit exists and what its value is.

Left-Hand and Right-Hand Limits

Left-hand and right-hand limits refer to the values that a function approaches as the input approaches a specific point from the left or right, respectively. For the limit of f(x) as x approaches -1, we must calculate the left-hand limit (using x < -1) and the right-hand limit (using x ≥ -1) to see if they are equal, which would indicate the overall limit exists.