Problem 36
Question
Finding a Limit In Exercises \(19-38,\) find the limit. $$ \lim _{x \rightarrow \infty} \cos \frac{1}{x} $$
Step-by-Step Solution
Verified Answer
So, the limit of \(\cos \frac{1}{x}\) as \(x\) approaches \(\infty\) is 1.
1Step 1: Understand the function
In the function \(\cos \frac{1}{x}\), as \(x\) gets larger and larger approaching \(\infty\), \(\frac{1}{x}\) gets closer and closer to 0.
2Step 2: Evaluate the limit
We know that \(\cos 0 = 1\). Therefore, as \(\frac{1}{x}\) gets closer and closer to 0, \(\cos \frac{1}{x}\) will get closer and closer to \(\cos 0\).
Key Concepts
Limits at InfinityTrigonometric LimitsFunctions and Limits
Limits at Infinity
Limits at infinity help us understand the behavior of functions as the independent variable, usually denoted as \(x\), grows very large or very negative. In our original problem, we have the limit \(\lim_{x \rightarrow \infty} \cos \frac{1}{x}\). Here, the term \(\frac{1}{x}\) is key. As \(x\) approaches infinity, \(\frac{1}{x}\) becomes smaller and smaller, effectively approaching zero. Consequently, the expression \(\cos \frac{1}{x}\) approaches \(\cos 0\), because the angle becomes tinier, trending towards zero.
Understanding limits at infinity allows us to determine that the value the function approaches can often be quite simple. In this case, the function approaches \(1\), which is \(\cos 0\). When dealing with limits at infinity it's important to evaluate the limiting behavior of the input variable. This means understanding how the input \(x\) affects expressions like \(\frac{1}{x}\) as \(x\) becomes infinitely large.
Understanding limits at infinity allows us to determine that the value the function approaches can often be quite simple. In this case, the function approaches \(1\), which is \(\cos 0\). When dealing with limits at infinity it's important to evaluate the limiting behavior of the input variable. This means understanding how the input \(x\) affects expressions like \(\frac{1}{x}\) as \(x\) becomes infinitely large.
Trigonometric Limits
Trigonometric limits involve understanding how trigonometric functions behave as their inputs approach certain values. Key to this understanding are the fundamental properties of trigonometric functions, such as periodicity and boundedness.
Let's take our example from the exercise: \(\lim_{x \rightarrow \infty} \cos \frac{1}{x}\). Relevant here is knowing that the cosine function oscillates between \(-1\) and \(1\). However, as \(x\) increases into infinity and \(\frac{1}{x}\) trends towards zero, the angle provided to the cosine function shrinks toward zero. The cosine of this nearing-zero angle stabilizes at \(1\), since \(\cos 0 = 1\).
Let's take our example from the exercise: \(\lim_{x \rightarrow \infty} \cos \frac{1}{x}\). Relevant here is knowing that the cosine function oscillates between \(-1\) and \(1\). However, as \(x\) increases into infinity and \(\frac{1}{x}\) trends towards zero, the angle provided to the cosine function shrinks toward zero. The cosine of this nearing-zero angle stabilizes at \(1\), since \(\cos 0 = 1\).
- Much of trigonometric limits' evaluations will center on simplifying expressions to basic angles where cosine or sine yields known results like \(0\), \(1\), or \(-1\).
- This understanding hinges on recognizing how angles change with respect to the input variable, and how trigonometric functions respond to those angles.
Functions and Limits
Functions and limits are fundamental in calculus, describing how a function behaves as the input approaches a specific value or grows without bound. In our exercise, the function of interest is \(\cos \frac{1}{x}\). The limit as \(x\) approaches infinity shows us how this specific composition behaves under extreme conditions.
The crux of evaluating functions and their limits lies in assessing the expressions within the function. As demonstrated, the key was simplifying \(\frac{1}{x}\) for large \(x\), understanding it approaches zero. This simplification directs us to a more manageable computation of the limit, showing that the function's outcome approaches a specific value—in this case, \(1\).
The crux of evaluating functions and their limits lies in assessing the expressions within the function. As demonstrated, the key was simplifying \(\frac{1}{x}\) for large \(x\), understanding it approaches zero. This simplification directs us to a more manageable computation of the limit, showing that the function's outcome approaches a specific value—in this case, \(1\).
- Recognizing how to break down complex expressions within functions can help tackle seemingly intricate limit problems.
- Thus, grasping functions and limits means being able to dissect and analyze interactions within the expressions as they approach pertinent values.
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