Problem 22 - Single Variable Calculus

Question

Sketch the graph of the function \(f\) and evaluate (a) \(\lim _{x \rightarrow a^{-}} f(x)\), (b) \(\lim _{x \rightarrow a^{+}} f(x)\), and (c) \(\lim _{x \rightarrow a} f(x)\) for the given value of a. \(f(x)=\left\\{\begin{array}{ll}x^{2}-1 & \text { if } x<1 \\ 2 & \text { if } x=1 ; \quad a=1 \\ \ln x & \text { if } x>1\end{array}\right.\)

Step-by-Step Solution

Answer

The graph of the given function can be represented by combining the functions \(f(x) = x^2 - 1\) for \(x < 1\), \(f(x) = 2\) for \(x = 1\), and \(f(x) = \ln x\) for \(x > 1\). Evaluating the given limits, \(\lim _{x \rightarrow 1^{-}} f(x) = 0\), \(\lim _{x \rightarrow 1^{+}} f(x) = 0\), and \(\lim _{x \rightarrow 1} f(x) = 0\). This indicates that the function \(f(x)\) is continuous at \(a = 1\).

Step 1: Sketch the graph

First, we will sketch the given functions on separate graphs and observe the points at which we need to change the functions for the given values of x. 1. For \(x<1\), the function is \(f(x) = x^2 - 1\). Plot this function for \(x<1\). 2. For \(x=1\), the function is \(f(x) = 2\). Plot only this point on the same graph as the previous function. 3. For \(x>1\), the function is \(f(x)=\ln x\). Plot this function for all values from \(x>1\). Combine these three graphs into one to represent the complete function \(f(x)\).

Step 2: Evaluate the limits from the left (\(a^-\)), from the right (\(a^+\)), and at the value \(a=1\).

(a) \(\lim _{x \rightarrow a^{-}} f(x)\): To evaluate this limit, we'd need to find the function \(f(x)\) as \(x\) approaches \(1\) from the left. Since \(f(x) = x^2 - 1\) when \(x < 1\), we use this expression for obtaining the left-hand limit: \(\lim _{x \rightarrow 1^{-}} f(x)=\lim_{x \rightarrow 1}(x^2 - 1) = (1)^2 - 1 = 0\) (b) \(\lim _{x \rightarrow a^{+}} f(x)\): To evaluate this limit, find the function \(f(x)\) as \(x\) approaches \(1\) from the right. As \(f(x) = \ln x\) when \(x > 1\), we use this expression for obtaining the right-hand limit: \(\lim _{x \rightarrow 1^{+}} f(x) = \lim_{x \rightarrow 1}(\ln x) = \ln{(1)} = 0\) (c) \(\lim _{x \rightarrow a} f(x)\): To evaluate the limit at \(a=1\), we must check whether the limits obtained in parts (a) and (b) are equal (\(\lim _{x \rightarrow a^{-}} f(x) = \lim _{x \rightarrow a^{+}} f(x)\)). As both left and right-hand limits are 0, we can conclude that: \(\lim _{x \rightarrow a} f(x) = \lim _{x \rightarrow 1} f(x) = 0\) Thus, the function \(f(x)\) is continuous at \(a=1\), and the corresponding left, right, and overall limits are all equal to 0.

Key Concepts

Limits of a Function

The concept of a limit in calculus is fundamental for understanding various behaviors of functions, especially as they approach a specific value or point. A limit tries to find out what value a function approaches as the input (or x-value) gets closer to a particular number. It doesn't necessarily matter what the function's value is at the point, but what it's approaching. In our exercise, evaluating \[lim _{x \rightarrow a} f(x)\] is essential to understand how the piecewise function behaves around the point a = 1.

How to Evaluate Limits

To evaluate limits, we often substitute the approached point into the function, if the function is continuous at the point. If direct substitution isn't possible, because the function doesn't exist at the point or is not continuous, other methods such as factorization, rationalization, or using special limits for trigonometric or logarithmic functions may be employed. In the given piecewise function, we use direct substitution for the left and right-hand limits as the functions x^2 - 1 and \[\ln x\] are continuous at their respective intervals that include the approach to x = 1.

Continuity in Calculus

A function is continuous at a point if there is no interruption in the graph of the function at that point. This means you could essentially draw the function at this point without lifting your pen from the paper. In formal terms, a function f(x) is continuous at a point a if and only if three conditions are met:

• The function is defined at a (i.e., f(a) exists).
• The limit of the function as x approaches a exists.
• The limit of the function as x approaches a is equal to the function's value at a (i.e., \[\lim_{x \rightarrow a} f(x) = f(a)\]).

For the provided piecewise function, we explore its continuity at a = 1. Since the limits from both sides of a are equal and the function is defined at that point, the function matches the criteria for continuity at that point in its domain.

Left-Hand Limit

When discussing limits, we often need to tackle the concept of the left-hand limit. It's about what happens to the function as the x-values approach a certain number from the left side, or, in other words, from lesser values.

Visualizing Left-Hand Limits

Imagine moving along the graph towards the point of interest from the left; this helps in understanding the behavior of the function as it approaches the point. Mathematically, it is represented as \[\lim_{x \rightarrow a^{-}} f(x)\], where a^{-} indicates we're approaching from values smaller than a. In our exercise, as x approaches 1 from the left, we examine the portion of the piecewise function x^2 - 1 to find that the left-hand limit is 0.

Right-Hand Limit

Complementing the left-hand limit is the right-hand limit, where we explore the behavior as x approaches a value from the right, meaning from greater values.

Analyzing Right-Hand Limits

When graphing, you'll move towards the point from the right to see what value the function is tending towards. Mathematically, we denote this as \[\lim_{x \rightarrow a^{+}} f(x)\], and the a^{+} symbol shows that the approach is from greater values. In the case of our function, as x approaches 1 from the right, we consider the \[\ln x\] part and determine the right-hand limit is also 0, matching the left-hand limit.