Problem 25 - Technical Mathematics with Calculus

Question

Limits Depending on Direction of Approach $$\lim _{x \rightarrow 0^{+}} \frac{x+1}{x}$$

Step-by-Step Solution

Answer

\(\lim_{x \to 0^{+}} \frac{x+1}{x} = +\infty\)

Step 1: Understand the Direction of the Limit

The notation \(\lim_{x \to 0^{+}} \frac{x+1}{x}\) indicates that we are looking at the limit as \(x\) approaches 0 from the positive side. This is called a one-sided limit, specifically a right-hand limit.

Step 2: Simplify the Expression

Since the function \(\frac{x+1}{x}\) is defined for \(x > 0\), we can split the fraction into two parts: \(\frac{x+1}{x} = \frac{x}{x} + \frac{1}{x} = 1 + \frac{1}{x}\).

Step 3: Evaluate the Limit

As \(x\) approaches 0 from the positive side, \(1 \to 1\) and \(\frac{1}{x}\) grows without bound. Therefore, the expression \(1 + \frac{1}{x}\) grows without bound as we get closer to 0 from the right. Thus, \(\lim_{x \to 0^{+}} \frac{x+1}{x} = +\infty\).

Key Concepts

Limits in Calculus

In the world of calculus, limits help us understand the behavior of functions as they approach a certain point or infinity. A limit can describe the value that a function approaches as the input (or 'x' value) gets closer to some number. For instance, when we look at the function \(f(x) = (x+1)/x\) as \(x\) approaches 0, we're asking, 'What value does \(f(x)\) get close to as \(x\) becomes very small?' However, limits are not always about finding a numerical value; they may also show us that a function is heading towards infinity or that it does not settle on a single value. This idea is at the heart of calculus because it defines continuity, rates of change, and the very slopes of curves at any point.

Right-Hand Limit

The right-hand limit is a specific type of limit where we only consider values of \(x\) that are greater than the point we're approaching, symbolized as \(x \to c^+\). It gives us a one-sided viewpoint of the function's behavior as \(x\) approaches a number from the positive side. Imagine walking along a path towards a sign, but you can only approach it from the right side. You're seeing how things look as you get closer, without crossing to the other side. Just like our exercise, \(\lim _{x \rightarrow 0^{+}} \frac{x+1}{x}\), where we are considering how the function behaves as we approach 0 from values greater than 0, like 0.1, 0.01, 0.001, and so on.

Asymptotic Behavior

Asymptotic behavior in mathematics refers to how a function behaves as it approaches a certain line or curve called an 'asymptote.' An asymptote can be vertical, horizontal, or even oblique, and it describes a boundary that the function will get infinitesimally close to but never actually reach. In our example, when we examine \(\lim _{x \rightarrow 0^{+}} \frac{x+1}{x}\), the function \(\frac{x+1}{x}\) demonstrates asymptotic behavior as it grows without bounds or, in mathematical terms, it approaches positive infinity. This vertical asymptote at \(x = 0\) indicates that no matter how close \(x\) gets to zero from the right, the function will increase towards infinity and never settle at a finite value.

Limit Calculations

Limit calculations require a mix of analytical and algebraic skills. It's about simplifying expressions and evaluating the behavior of functions as they near certain values. As demonstrated in the solution process of our exercise, we first simplified the function by splitting the fraction and then assessed how each part behaves as \(x\) approaches 0 from the right. The term \(1/x\) is crucial because it tells us that the function's value gets larger and larger as \(x\) gets smaller and smaller. More formally, this approach of breaking down the function and evaluating each term's limit separately allows us to determine that the limit of \(\frac{x+1}{x}\) as \(x\) goes to 0 from the positive side is positive infinity. Recall that in limit calculations, not every expression will lead to a straightforward finite number; some will shoot off to infinity or indicate that the limit does not exist in a conventional sense.