Question

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 of the function \(x^{2}+1\) as \(x\) approaches 1 is \(2\). The \(\varepsilon-\delta\) proof verifies this result.

Step-by-step solution

Find the Limit

We can directly substitute \(x = 1\) into the function \(x^{2}+1\) to obtain \(L = 1^{2} + 1 = 2\)

Define Function and Limit

We establish the function for which we want to verify the limit: \(f(x)=x^{2}+1\) and we have found the limit \(L = 2\) in step 1.

Start \(\varepsilon - \delta\) proof

The goal is to make \(|f(x) - L|<\varepsilon\). We have: \(|f(x) - L| = |x^{2} +1 - 2| = |x-1||x+1|\). We will use the fact \(|x+1| ≤ 3\) which is valid whenever \(|x-1|<1\). Thus if we choose \(\delta = \min(1, \frac{\varepsilon}{3})\), we ensure that if \(|x-1|<\delta\), then \(|x+1| < 3\) and \(|x-1|<\frac{\varepsilon}{3}\). Therefore, \( |f(x) - L| = |x-1||x+1| < 3|x-1| < 3\frac{\varepsilon}{3} = \varepsilon \). This completes the proof.

Key concepts

Limits in Calculus

Limits are fundamental to calculus; they describe the behavior of functions as they approach specific points or infinity. When calculating a limit, we're often interested in the value a function approaches as the input gets closer to a certain number. For instance, the expression \( \lim_{x \rightarrow a}f(x) \) is read as 'the limit of \(f(x)\) as \(x\) approaches \(a\).' In our example, we seek to find \( \lim_{x \rightarrow 1}(x^2+1) \) which helps investigate the behavior of the function near \(x = 1\).

Understanding limits is not just about plugging in values; it's about grasping the concept of approaching. Even when direct substitution isn't possible, especially when it results in an indeterminate form like \(0/0\), other techniques such as factoring, rationalization, or L'Hopital's rule might be employed. Limits are the groundwork for continuity, derivatives, and integration, which are the pillars of calculus.

Continuity of Functions

A function is said to be continuous at a point if it meets three criteria: the function is defined at that point, the limit exists at that point, and the limit equals the function's value at that point. In mathematical terms, \(f\) is continuous at \(a\) if \( \lim_{x \rightarrow a}f(x) = f(a)\).

In our example, the function \(f(x) = x^2 + 1\) is continuous at \(x = 1\), as we found the limit to be 2, which is the same as the function's value when \(x\) is exactly 1. Continuity ensures that there are no breaks, jumps, or holes at that point in the function, making the graph of the function smooth and unbroken around that area. This property is crucial for predicting behavior within a certain range without evaluating every single point.

Calculus Proofs

Proving mathematical statements, like the validity of a limit, employs logical reasoning to demonstrate the truthfulness of the assertion. \(\varepsilon-\delta\) proofs embody this rigor in calculus; they provide a formal method for verifying limits.

Structurally, an \(\varepsilon-\delta\) proof consists of several stages. Initially, it's important to express what you're trying to prove. In our exercise, we wanted to prove that \( \lim_{x \rightarrow 1}(x^2+1) = 2\). Next, the proof constructs an argument around the concept of making the difference \( |f(x) - L|\) arbitrarily small (less than any positive number \(\varepsilon\)) by choosing \(x\) sufficiently close to the limit point (within a distance \(\delta\) from it). If this condition can be satisfied for any \(\varepsilon > 0\) by finding a corresponding \(\delta\), the limit is proved. This logical progression ensures a sound and universally accepted proof.

Limit Laws

Limit laws are rules that simplify the process of finding limits and add structure to the way limits behave. They allow us to break down complex expressions into simpler parts, making calculations more manageable. Some of the basic limit laws include the sum law, product law, and quotient law, which respectively allow splitting sums, products, and quotients of functions into individual limits.

Applying these laws, for instance, the limit of a sum can be found by taking the limit of each addend separately. The exercise we considered deals with a polynomial, and using these laws, we can evaluate the limit of a polynomial term-by-term if needed. The elegance of these laws lies in their ability to streamline the process, often enabling us to foresee the limit of a function without detailed calculations. However, it's crucial to remember when these laws are applicable and when caution is needed, such as with indeterminate forms.