Problem 36 - Calculus Early Transcendentals

Question

If \(2 x \leqslant g(x) \leqslant x^{4}-x^{2}+2\) for all \(x,\) evaluate \(\lim _{x \rightarrow 1} g(x)\)

Step-by-Step Solution

Answer

The limit of \(g(x)\) as \(x\) approaches 1 is 2.

Step 1: Express the Inequality Constraints

The problem states that for all \(x\), \(2x \leq g(x) \leq x^4 - x^2 + 2\). We want to evaluate the limit of \(g(x)\) as \(x\) approaches \(1\).

Step 2: Apply the Limit to Each Part of the Inequality

To find \(\lim_{{x \to 1}} g(x)\), we consider the limits of the boundary functions: \(2x\) and \(x^4 - x^2 + 2\) as \(x\) approaches \(1\).

Step 3: Calculate the Left Boundary Limit

Calculate \(\lim_{{x \to 1}} 2x\): Since \(2x\) is continuous, \(\lim_{{x \to 1}} 2x = 2 \cdot 1 = 2\).

Step 4: Calculate the Right Boundary Limit

For the right boundary, compute \(\lim_{{x \to 1}} (x^4 - x^2 + 2)\). Substituting \(x = 1\), we get \(1^4 - 1^2 + 2 = 2\).

Step 5: Apply the Squeeze Theorem

Since \(2x \leq g(x) \leq x^4 - x^2 + 2\) for all \(x\), and both bounds approach 2 as \(x\) approaches 1, by the Squeeze Theorem, \(\lim_{{x \to 1}} g(x) = 2\).

Key Concepts

Squeeze Theorem

The Squeeze Theorem is a valuable tool in calculus for finding the limit of a function when it is difficult to compute directly. It is particularly useful when a function is "trapped" between two other functions whose limits are known or easier to determine.
The theorem works by "squeezing" the function of interest between two simpler functions. If the upper and lower functions converge to the same limit at a certain point, then the function squeezed between them must also converge to that same limit.

• First, you identify two functions, say \( f(x) \) and \( h(x) \), such that \( f(x) \leq g(x) \leq h(x) \) over an interval.
• Next, compute \( \lim_{{x \to a}} f(x) \) and \( \lim_{{x \to a}} h(x) \).
• If both limits are equal to a number \( L \), then \( \lim_{{x \to a}} g(x) = L \).

To apply the Squeeze Theorem successfully, make sure the inequality \( f(x) \leq g(x) \leq h(x) \) holds for values of \( x \) close to \( a \), except possibly at \( a \) itself.

Inequalities in Calculus

Understanding and working with inequalities is essential in calculus. Inequalities allow us to estimate and bound the behavior of functions, providing crucial insights into their properties.
In the context of limits, inequalities can help us establish upper and lower bounds for functions. This is especially useful when applying the Squeeze Theorem or similar techniques.

• Inequalities can inform us about the continuity and boundedness of a function.
• We often express functions in terms of inequalities to simplify the evaluation of a limit.
• Understanding the interplay between functions using inequalities helps in establishing crucial relationships, such as dominance or convergence, as \( x \) approaches a particular value.

When dealing with inequalities, always check whether they remain valid over the interval of interest. These tools serve to create a bridge between complex expressions and simpler evaluations, leading to a greater understanding of the function's behavior.

Continuity of Functions

Continuity is a fundamental property in calculus that implies a function behaves predictably and smoothly without any jumps or interruptions.
A function is continuous at a point \( x = a \) if three conditions are satisfied:

• The function \( f(x) \) is defined at \( a \).
• The limit \( \lim_{{x \to a}} f(x) \) exists.
• The limit and the function value are equal, i.e., \( \lim_{{x \to a}} f(x) = f(a) \).

Understanding and identifying continuity in functions is essential as it simplifies the process of finding limits. Continuous functions at a point or over an interval allow for straightforward limit calculations, where the limit can be evaluated by simple substitution.
Additionally, continuous functions are more predictable, which aids in applying theorems like the Intermediate Value Theorem, further emphasizing the importance of continuity in mathematical analysis and problem-solving.