Math Problem Statement

lim x->-3- ([x]/(sign(x+3)))

Solution

To evaluate the limit:

limx3xsign(x+3)\lim_{x \to -3^-} \frac{\lfloor x \rfloor}{\operatorname{sign}(x+3)}

we need to understand the behavior of the floor function x\lfloor x \rfloor and the sign function sign(x+3)\operatorname{sign}(x+3) as xx approaches 3-3 from the left.

  1. Analyzing sign(x+3)\operatorname{sign}(x+3):
    The function sign(x+3)\operatorname{sign}(x+3) gives the sign of x+3x + 3. Specifically:

    • If x>3x > -3, x+3x + 3 is positive, so sign(x+3)=1\operatorname{sign}(x + 3) = 1.
    • If x=3x = -3, x+3=0x + 3 = 0, but we don't evaluate the sign at this exact point due to the limit approaching from the left.
    • If x<3x < -3, x+3x + 3 is negative, so sign(x+3)=1\operatorname{sign}(x + 3) = -1.

    Since we're approaching 3-3 from the left (x3x \to -3^-), x<3x < -3 in this context, and thus sign(x+3)=1\operatorname{sign}(x+3) = -1.

  2. Analyzing x\lfloor x \rfloor: The floor function x\lfloor x \rfloor takes the greatest integer less than or equal to xx.

    • As x3x \to -3^-, values of xx just to the left of 3-3 (like 3.1-3.1, 3.01-3.01, etc.) would have x=4\lfloor x \rfloor = -4.
  3. Substitute and Simplify: Given these behaviors, we have: xsign(x+3)=41=4\frac{\lfloor x \rfloor}{\operatorname{sign}(x+3)} = \frac{-4}{-1} = 4

Therefore, the limit is:

limx3xsign(x+3)=4\lim_{x \to -3^-} \frac{\lfloor x \rfloor}{\operatorname{sign}(x+3)} = 4


Would you like further clarification on any part of this? Here are some related questions you might explore:

  1. What is the limit if xx approaches 3-3 from the right?
  2. How does the behavior of the floor function affect limits near integers?
  3. What are other examples where the sign function is used in limits?
  4. How would this limit change if we used the ceiling function instead of the floor function?
  5. Why is the left-hand limit different from a right-hand limit in piecewise functions?

Tip: In limits involving floor functions, consider the integer behavior of x\lfloor x \rfloor around the point of interest—approaching from left or right can yield different integer values.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Limits
Floor Function
Sign Function
Left-Hand Limit

Formulas

\lim_{x \to a} \frac{f(x)}{g(x)}
floor function \(\lfloor x \rfloor\)
sign function \(\operatorname{sign}(x)\)

Theorems

Limit of a function
Piecewise behavior of floor and sign functions

Suitable Grade Level

Grades 11-12