Question

Using the \(\varepsilon-\delta\) Definition of Limit In Exercises \(37-48\) , find the limit \(L\) . Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L .\) $$ \lim _{x \rightarrow 1}\left(x^{2}+1\right) $$

Short answer

The limit \(L\) of the function \(x^{2}+1\) as \(x\) approaches \(1\) is \(2\).

Step-by-step solution

Find the limit \(L\)

We substitute \(x = 1\) into \(x^{2}+1\) to find the limit 'L'. Now \(L = (1)^{2} + 1\) which simplifies to \(L = 2\)

Use the \(\varepsilon-\delta\) definition

Now use the \(\varepsilon-\delta\) definition to prove that the limit is 'L'. As per the \(\varepsilon-\delta\) definition, for each \(\varepsilon > 0\), there must be a \(\delta > 0\) such that whenever \(0< |x-1| < \delta \), it follows that \(|x^{2} + 1 - 2| < \varepsilon\). This can be simplified to \( |x - 1||x + 1| < \varepsilon\). Since x is approaching 1, we can choose \(\delta = \varepsilon\) assuming that \(\varepsilon <= 1\). If \(\varepsilon > 1\), we can simply choose \(\delta = 1\), in which case \(|x -1| <= 1\) and therefore \(|x + 1| <= 3\). Hence the initial equation holds and we have proven the limit according to the \(\varepsilon-\delta\) definition.

Key concepts

Limits

Limits help us understand the behavior of a function as it approaches a particular point. In simple terms, when we say \( \lim_{x \rightarrow 1} (x^2 + 1) = 2 \), we mean that as \( x \) gets closer and closer to \( 1 \), the value of the expression \( x^2 + 1 \) gets closer to \( 2 \).
This is a crucial concept in calculus, as it helps set the foundation for more complex ideas, like derivatives and integrals. To find the limit of a function, we substitute the value where \( x \) is approaching into the function. If the function is continuous at this point, the limit will equal the value of the function. For example, substituting \( x = 1 \) into \( x^2 + 1 \) yields \( 2 \). Hence the limit \( L \) is \( 2 \).
Limits are fundamental in defining instantaneous rates of change, which are key in calculus.

Calculus

Calculus is a branch of mathematics focusing on limits, functions, derivatives, integrals, and infinite series. It provides tools for modeling situations where quantities change, which makes it invaluable in physics, engineering, economics, and beyond.
One of the primary purposes of calculus is to study changes and motion. For this, it uses concepts like limits to analyze how a function behaves as its input approaches some point. This helps elucidate dynamic systems, calculate slopes of curves, and find areas under curves. By establishing a groundwork for differential calculus (concerned with rates of change) and integral calculus (concerned with accumulation of quantities), we can solve problems like finding the speed of a moving car or forecasting trends.
Calculus continues to be a vital tool in scientific and engineering fields due to its ability to model and solve real-world problems.

Proof by Epsilon-Delta

Proof by epsilon-delta is a formal way of proving limits in calculus. This method precisely shows that as the input of a function approaches a certain point, the output approaches a specific value. It's a rigorous way to demonstrate the existence and correctness of limits.
The epsilon-delta definition states: For every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < |x - c| < \delta\), then \(|f(x) - L| < \varepsilon\). In our example, where \( \lim_{x \rightarrow 1} (x^2 + 1) = 2 \), this definition works by choosing \(\delta\) cleverly or carefully depending on the given \(\varepsilon\).
This method shows how precise mathematics can get when defining and proving properties of functions, ensuring no room for ambiguity.

Mathematical Proofs

Mathematical proofs are logical arguments verifying the truths of mathematical statements. Proofs use previously established facts and logical reasoning to show that a statement is unequivocally true.
In calculus, and specifically with limits, proofs like the epsilon-delta proof are used for clarity and verification. By using precise definitions and logical steps, these proofs ensure that results are correct and understandable by others in the mathematical community. They are a crucial part of mathematical communication, helping to establish trust and accuracy in mathematical results.
Proof techniques can vary widely. Some may rely on algebraic manipulation, others on geometric interpretation, and some, like the epsilon-delta method, on a mixture of both logical and mathematical manipulation. They are fundamental to advancing knowledge and understanding in mathematics and its applications.