Question

Find the limit of the function f(x) as x approaches -1, where f(x) is defined as follows: f(x) = 2x if x ≥ -1 and f(x) = x² + 3 if x < -1.

Answer 1

As x approaches -1 from the right, the function f(x) approaches -2

As x approaches -1 from the left, the function f(x) approaches 4

The limit of the function f(x) as x approaches -1 is undefined

Step-by-step explanation:

Evaluate the function from both sides of -1 and see if the values converge to a specific number.

x is greater than or equal to -1: the function f(x) is defined as 2x. As x approaches -1 from the right side (x > -1), the value of 2x approaches 2*(-1) = -2.

x is less than -1: the function f(x) is defined as x² + 3. As x approaches -1 from the left side (x < -1), the value of x² + 3 approaches (-1)² + 3 = 4.

Answer 2

To find the limit of the function f(x) as x approaches -1, we need to determine the value that f(x) approaches as x gets closer and closer to -1 from both sides. Since f(x) is defined as 2x for x ≥ -1 and x² + 3 for x < -1, we need to evaluate these two expressions separately. The limit of f(x) as x approaches -1 is undefined.

Step-by-step explanation:

To find the limit of the function f(x) as x approaches -1, we need to determine the value that f(x) approaches as x gets closer and closer to -1 from both sides. Since f(x) is defined as 2x for x ≥ -1 and x² + 3 for x < -1, we need to evaluate these two expressions separately.

When x ≥ -1, f(x) = 2x. So as x approaches -1 from the right side (x > -1), the function approaches 2(-1) = -2.

When x < -1, f(x) = x² + 3. As x approaches -1 from the left side (x < -1), f(x) approaches (-1)² + 3 = 4.

Since the function approaches -2 from the right side and 4 from the left side, the limit of f(x) as x approaches -1 is undefined.

To find the limit of the function f(x) as x approaches -1, where f(x) = 2x if x ≥ -1 and f(x) = x² + 3 if x < -1, we consider the left-hand limit and the right-hand limit separately.

For the right-hand limit as x approaches -1 from the right (x ≥ -1), f(x) is defined as 2x. So,

lim_(x → -1^+) f(x) = lim_(x → -1^+) 2x = 2(-1) = -2.

For the left-hand limit as x approaches -1 from the left (x < -1), f(x) is defined as x² + 3. So,

lim_(x → -1^-) f(x) = lim_(x → -1^-) x² + 3 = (-1)² + 3 = 1 + 3 = 4.

Since the left-hand limit and right-hand limit as x approaches -1 are not equal, we can conclude that the limit of f(x) as x approaches -1 does not exist.