Math Problem Statement

solve

If 2x ≤ g(x) ≤ x^4 - x^2 + 2 for all x, evaluate lim(x→1) g(x).

Solution

We are tasked with finding limx1g(x)\lim_{x \to 1} g(x), given the inequality:

2xg(x)x4x2+2for allx.2x \leq g(x) \leq x^4 - x^2 + 2 \quad \text{for all} \quad x.

Step 1: Analyzing the limit using the Squeeze Theorem

The Squeeze Theorem can be applied here because g(x)g(x) is squeezed between two functions, 2x2x and x4x2+2x^4 - x^2 + 2.

We will evaluate the limits of these bounding functions as x1x \to 1:

Lower Bound: limx12x\lim_{x \to 1} 2x

limx12x=2(1)=2.\lim_{x \to 1} 2x = 2(1) = 2.

Upper Bound: limx1x4x2+2\lim_{x \to 1} x^4 - x^2 + 2

limx1(x4x2+2)=(1412+2)=11+2=2.\lim_{x \to 1} (x^4 - x^2 + 2) = (1^4 - 1^2 + 2) = 1 - 1 + 2 = 2.

Step 2: Applying the Squeeze Theorem

Since both the lower bound and upper bound approach 2 as x1x \to 1, by the Squeeze Theorem, the limit of g(x)g(x) must also approach 2:

limx1g(x)=2.\lim_{x \to 1} g(x) = 2.

Final Answer:

limx1g(x)=2.\lim_{x \to 1} g(x) = 2.

Would you like further details or clarifications?

Here are 5 related questions to deepen your understanding:

  1. How does the Squeeze Theorem work in other cases?
  2. What are the conditions for applying the Squeeze Theorem?
  3. Can g(x)g(x) be uniquely determined from the given inequality?
  4. What is the significance of limit evaluation in calculus?
  5. How would the solution change if the bounds for g(x)g(x) were different?

Tip: Always check that the limits of the bounding functions converge to the same value when using the Squeeze Theorem.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Limits
Inequalities
Squeeze Theorem

Formulas

lim(x→1) 2x = 2
lim(x→1) (x^4 - x^2 + 2) = 2

Theorems

Squeeze Theorem

Suitable Grade Level

Grades 11-12