Problem 10 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises 9-12, find the limit and confirm your answer using the Sandwich Theorem. $$\lim _{x \rightarrow-\infty} \frac{1-\cos x}{x^{2}}$$

Step-by-Step Solution

Answer

The limit of the function \(\frac{1-\cos x}{x^{2}}\) as \(x\) approaches negative infinity is \(0\).

Step 1: Finding Limit

Firstly, the limit of the given function should be found as \(x\) approaches negative infinity. Due to the complexity of the function, it should be split into two parts to ease calculation:\[\lim_{x \rightarrow-\infty} \frac{1}{x^{2}}- \lim_{x \rightarrow-\infty}\frac{\cos x}{x^{2}}.\]The limit of \(\frac{1}{x^{2}}\) as \(x\) approaches negative infinity is zero. The value of cosine doesn't depend on \(x\) and since \(x\) approaches negative infinity, \(\frac{\cos x}{x^{2}}\) will also tend to zero. This gives limit as \(0-0=0\).

Step 2: Confirming using Sandwich Theorem

The Sandwich Theorem can now be utilized to verify the obtained answer. It should be noted that \(-1 \leq \cos x \leq 1\). As such, for the negative fraction, the lower limit will be \(\frac{1-1}{x^2}=0\) and the upper limit will be \(\frac{1+1}{x^2}= \frac{2}{x^{2}}\).As \(x\) approaches negative infinity, both the lower and the upper boundaries will tend to zero as well, confirming that the original function must approach zero too, as it is 'sandwiched' between these two values.

Key Concepts

Sandwich Theorem

The Sandwich Theorem is a powerful tool in calculus, especially when dealing with limits. It helps find the limit of a function that is "sandwiched" between two other functions. If you can show that two functions both approach the same limit, then any function trapped between them will approach that limit too.
In this exercise, we used the properties of the cosine function, where

\(-1 \leq \cos x \leq 1\)

to create bounds for the function \(\frac{1-\cos x}{x^2}\). The upper bound was \(\frac{2}{x^2}\) and the lower \(\frac{0}{x^2}\), both tending to zero as \(x\) approaches negative infinity.

• It builds on simple observations about the range of values \(\cos x\) can take.
• Useful in showing limits when direct calculation is complex or impossible.

An important tool in many calculus problems, helping to confirm or find limits.

Infinite Limits

Infinite limits explore what happens to a function as the variable goes towards infinity or negative infinity. In this context, we're interested in how the terms \(\frac{1}{x^2}\) and \(\frac{\cos x}{x^2}\) behave when \(x\) approaches negative infinity.
The first term \(\frac{1}{x^2}\) clearly approaches zero, as any positive number divided by an increasingly large number gets smaller and smaller.

• This idea helps simplify many complex expressions.
• Negative infinity just means we're moving further left on the number line—it's not inherently different from positive infinity in these calculations.

It's key to remember that both terms tend toward zero, driving the limit of the whole expression to zero.

Cosine Function

The cosine function, \(\cos x\), is periodic with a range of values between -1 and 1. This property is crucial when applying the Sandwich Theorem.
The periodicity implies that \(\cos x\) doesn't grow or shrink infinitely. Instead, it repeats every cycle.

• This repetition is part of why we can define limits like in the exercise.
• Understanding this aids in visualizing why it doesn't impact the limit as \(x\) approaches infinity.

When the rest of the function's terms handle the convergence, cosine's behavior ensures it doesn't upset the balance.

Calculus Problem Solving

Calculus problem solving often involves breaking down complex problems into simpler pieces. This exercise was initially complicated but splitting it into manageable parts made it easier:
We started by analyzing individual terms. These small steps were key to finding the overall limit.

• Earlier steps simplify how each part behaves at infinity.
• Combining insights from different limits confirms the solution systematically.

These methods are common in calculus. They show how foundational tools, like limit laws and inequalities, apply effectively in different contexts.