Problem 10 - Thomas Calculus

Question

If \(f(1)=5,\) must \(\lim _{x \rightarrow 1} f(x)\) exist? If it does, then must \(\lim _{x \rightarrow 1} f(x)=5 ?\) Can we conclude anything about lim \(_{x \rightarrow 1} f(x) ?\) Explain.

Step-by-Step Solution

Answer

The existence of \( \lim _{x \rightarrow 1} f(x) \) cannot be concluded just from \( f(1)=5 \); its value may not be 5.

Step 1: Understanding the Limit Definition

The limit \( \lim _{x \rightarrow a} f(x) \) exists if as \( x \) approaches \( a \), the values of \( f(x) \) approach a particular single real number \( L \). This means that to conclude the limit exists, \( f(x) \) should be approaching the same value from both the left and right as \( x \) approaches \( a \). In this case, as \( x \rightarrow 1 \).

Step 2: Check the Given Function Value

The problem states that \( f(1)=5 \). This tells us the exact value of the function at \( x=1 \), but it doesn't directly inform us about the behavior of the function as \( x \rightarrow 1 \). Thus, knowing the function's value at \( x=1 \) isn't sufficient to conclude the behavior around it.

Step 3: Determine if Limit Must Exist

\( \lim _{x \rightarrow 1} f(x) \) will exist if \( f(x) \), approaching from both directions (left and right) towards \( x=1 \), tends to a single value. Given only \( f(1)=5 \), we don't have enough information about \( f(x) \) to conclusively determine that the limit exists as \( x \rightarrow 1 \). The existence of a limit depends on the continuity or behavior of \( f(x) \) around \( x = 1 \), which isn't provided.

Step 4: Does Limit Equal the Function Value?

Even if the limit \( \lim _{x \rightarrow 1} f(x) \) does exist, it does not necessarily have to equal the function value \( f(1) = 5 \). For example, a function could approach 4 as \( x \rightarrow 1 \) but still have \( f(1) = 5 \). Without more information, we cannot assume \( \lim _{x \rightarrow 1} f(x) = 5 \).

Step 5: Conclusion on the Limit

Based on the given information, we cannot make any definitive conclusion about \( \lim _{x \rightarrow 1} f(x) \). We need additional data regarding the function's behavior as it approaches \( x=1 \) from both sides to determine both the existence of the limit and its value.

Key Concepts

Continuity

Continuity is like the smooth running of a movie. Just as each frame glides seamlessly into the next without any jumps, a function is continuous at a point if there are no interruptions or gaps in its behavior around that point. For a function to be continuous at a certain point, it needs to satisfy three conditions:

• The function is defined at the point.
• The limit of the function exists as it approaches that point.
• The limit equals the function value at that point.

If any of these conditions fail, the function has a discontinuity at that point. So, in our exercise, even though we know that
\( f(1) = 5 \), this only tells us that the function is defined at \( x = 1 \), but it doesn't inform us about the limit or whether the function is continuous as \( x \rightarrow 1 \).

Limit Definition

The concept of a limit captures the idea of a function approaching a particular value as the input gets closer and closer to a specific point. Let's break it down further:

• For \( \lim_{x \rightarrow a} f(x) \) to exist, the values of \( f(x) \) need to hone in on a single number \( L \) as \( x \) moves towards \( a \).
• This must happen regardless of whether \( x \) approaches \( a \) from the left or the right.

The definition of a limit doesn't require that the function \( f(x) \) equals \( L \) at \( x = a \). It purely describes the behavior of \( f(x) \) as it nears this point. In our exercise, knowing the value \( f(1) \) gives no direct indication about the values \( f(x) \) takes on as \( x \) approaches 1. Thus, the given information isn't sufficient to determine whether the limit exists.

Function Behavior

A function's behavior as it approaches a particular point can reveal much about the existence and nature of a limit. Here are some scenarios to consider:

• If \( f(x) \) approaches a single finite value from both directions as \( x \rightarrow 1 \), then the limit exists.
• Conversely, if \( f(x) \) behaves erratically or diverges from different sides, the limit does not exist.

Function behavior includes observing how \( f(x) \) changes as \( x \) gets closer to a point and can be influenced by sudden jumps, asymptotes, or oscillations.
In the context of our exercise, despite knowing \( f(1) = 5 \), we lack information about how \( f(x) \) behaves when nearing \( x = 1 \). Without insights into this approaching behavior, we cannot conclusively pinpoint the existence or value of the limit.