Math Problem Statement

Limite quand x tend vers 0

Limite quand x tend vers 0 pour f(x) = (3x^2 + 2x + 1) / (2x^2 + 4x)

Solution

To evaluate the limit of f(x)=3x2+2x+12x2+4xf(x) = \frac{3x^2 + 2x + 1}{2x^2 + 4x} as x0x \to 0, follow these steps:

  1. Substitute x=0x = 0 directly to see if the limit can be evaluated directly: f(x)=3(0)2+2(0)+12(0)2+4(0)=10.f(x) = \frac{3(0)^2 + 2(0) + 1}{2(0)^2 + 4(0)} = \frac{1}{0}. The denominator becomes zero, so the function is undefined at x=0x = 0, and we need further analysis.

  2. Factorize and simplify the expression:

    • Numerator: 3x2+2x+13x^2 + 2x + 1 has no immediate factorization relevant here.
    • Denominator: 2x2+4x=2x(x+2)2x^2 + 4x = 2x(x + 2).

    The function becomes: f(x)=3x2+2x+12x(x+2).f(x) = \frac{3x^2 + 2x + 1}{2x(x + 2)}.

  3. Consider the behavior near x=0x = 0:

    • The term 2x(x+2)2x(x + 2) in the denominator indicates that f(x)f(x) has a vertical asymptote at x=0x = 0.
  4. Left-hand and right-hand limits: Since 1x\frac{1}{x} appears in the function when expanded, the left-hand limit (x0x \to 0^-) and right-hand limit (x0+x \to 0^+) differ in sign (approaching -\infty and ++\infty, respectively).

Conclusion:

The limit does not exist because the left-hand and right-hand limits are not equal. The function exhibits a vertical asymptote at x=0x = 0.

Would you like a step-by-step explanation of vertical asymptotes or limits at infinity?


Related Questions:

  1. What are the vertical asymptotes of a rational function?
  2. How do you determine the behavior of a function near singularities?
  3. What happens to the numerator and denominator as x0x \to 0?
  4. Can the numerator or denominator be factored further?
  5. What tools can help analyze one-sided limits?

Tip:

Always check both one-sided limits when the denominator approaches zero to determine the behavior of the function!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Limits
Rational Functions
Asymptotes
Factoring

Formulas

Limit evaluation for rational functions
Factoring expressions

Theorems

Limit laws
Behavior of functions near singularities

Suitable Grade Level

Grades 11-12