Problem 102
Question
Proof Use the definition of infinite limits at infinity to prove that \(\lim _{x \rightarrow \infty} x^{3}=\infty\)
Step-by-Step Solution
Verified Answer
\(\lim_{x \rightarrow \infty} x^{3} = \infty\)
1Step 1: Understand the definition of infinite limits
The definition of infinite limits at infinity is as follows: We say that \(\lim_{x \rightarrow \infty} f(x) = \infty\) if for every positive number \(M\), there exists a number \(N\) such that if \(x > N\), then \(f(x) > M\).
2Step 2: Apply the definition to the given function
We want to prove that \(\lim_{x \rightarrow \infty} x^{3} = \infty\). This means we want to show that for every positive number \(M\), we can find a number \(N\) such that if \(x > N\), then \(x^{3} > M\). We can choose \(N = \sqrt[3]{M}\). Then, whenever \(x > N\), we have \(x > \sqrt[3]{M}\), thus \(x^{3} > (\sqrt[3]{M})^3 = M\).
3Step 3: Conclusion
So, we have shown that for every positive \(M\), there is a \(N = \sqrt[3]{M}\) such that if \(x > N\), then \(x^{3} > M\). Therefore, by the definition of infinity limit, \(\lim_{x \rightarrow \infty} x^{3} = \infty\).
Key Concepts
Infinite LimitsCalculusProofs in CalculusLimits and Continuity
Infinite Limits
When we delve into the world of calculus, the concept of 'infinite limits' is essential to mastering the subject. Picture a graph shooting upwards without bound as we move along the x-axis to the right—that's an infinite limit at play. To understand infinite limits formally, we consider that as the variable 'x' approaches infinity, the function 'f(x)' grows without bound. Students often visualize this as a line on a graph that continues to rise off into space.
In our exercise, we demonstrate this concept by proving that the limit as 'x' goes to infinity of 'x^3' is infinity. We use the definition which asserts that for any number 'M' that you can think of, there’s a point 'N' on the graph after which 'x^3' will be greater than 'M'. No matter how large 'M' is, 'x^3' can surpass it, showing the nature of limitless growth.
In our exercise, we demonstrate this concept by proving that the limit as 'x' goes to infinity of 'x^3' is infinity. We use the definition which asserts that for any number 'M' that you can think of, there’s a point 'N' on the graph after which 'x^3' will be greater than 'M'. No matter how large 'M' is, 'x^3' can surpass it, showing the nature of limitless growth.
Calculus
Calculus, at its heart, is the mathematics of change and motion. It helps us understand the behavior of functions and the way they evolve as their input changes. Infinite limits are just one example of the intriguing concepts that calculus introduces. Among other things, calculus allows us to compute areas, volumes, rates of change (derivatives), and accumulation of quantities (integrals).
It's the language of engineers, economists, physicists, and many other professionals who need to model and analyze real-world phenomena. As you study calculus, concepts like limits, derivatives, and integrals become tools that equip you to tackle complex problems with confidence.
It's the language of engineers, economists, physicists, and many other professionals who need to model and analyze real-world phenomena. As you study calculus, concepts like limits, derivatives, and integrals become tools that equip you to tackle complex problems with confidence.
Proofs in Calculus
Proofs are the bedrock of mathematics and calculus is no exception. A proof is a logical argument that establishes the truth of a statement beyond any doubt. It might sound intimidating at first, but proofs outline the reason why things are as we say they are in mathematics, thus ensuring that we're not just making educated guesses.
In our exercise, we use a 'direct proof' to verify that as 'x' increases without bound, so does 'x^3'. We argue that, by the definition, the function's value will exceed any fixed value 'M' after a certain point—we find an 'N' that anchors our argument. This systematic approach builds the certainty needed in mathematics, where accuracy is not just important; it’s everything.
In our exercise, we use a 'direct proof' to verify that as 'x' increases without bound, so does 'x^3'. We argue that, by the definition, the function's value will exceed any fixed value 'M' after a certain point—we find an 'N' that anchors our argument. This systematic approach builds the certainty needed in mathematics, where accuracy is not just important; it’s everything.
Limits and Continuity
The concepts of limits and continuity are intertwined in calculus. A limit studies the behavior of a function as the input approaches a particular value. When the function approaches a specific number as the input gets close to a point, we say the function has a 'limit' at that point. If the function is defined at that point and equal to the limit, then it's continuous there. Continuity means that the function doesn't have any abrupt breaks or jumps – it's smooth.
Continuity is critical because it guarantees that small changes in the input lead to small changes in the output, which is a reassuring property in both mathematics and the sciences. It tells us that our mathematical models behave predictably and consistently over the region of interest.
Continuity is critical because it guarantees that small changes in the input lead to small changes in the output, which is a reassuring property in both mathematics and the sciences. It tells us that our mathematical models behave predictably and consistently over the region of interest.
Other exercises in this chapter
Problem 100
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