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Chapter 2: Problem 76

In the following exercises, sketch the graph of a function with the given properties. $$ \lim _{x \rightarrow 2} f(x)=1, \lim _{x \rightarrow 4^{-}} f(x)=3, \quad \lim _{x \rightarrow 4^{+}} f(x)=6, x=4 $$ is not defined.

Short Answer

Expert verified
The function approaches y=1 at x=2, approaches 3 from the left and 6 from the right of x=4, but is not defined at x=4.

Step by step solution

01

Understanding Limits

The problem provides limits at certain points, which means the function approaches these values as x gets closer to those points. We specifically need to consider behavior as x approaches 2 and from both sides as x approaches 4.
02

Limit at x = 2

The limit \(\lim _{x \rightarrow 2} f(x)=1\)\ implies that as x gets closer to 2 from both sides, the function approaches 1. The graph will have a point at \(x = 2\) very close to the value 1, showing that \(f(x)\) tends towards 1.
03

Left-Hand Limit at x = 4

The left-hand limit \(\lim _{x \rightarrow 4^{-}} f(x)=3\)\ indicates that as x approaches 4 from values less than 4, the function approaches the value 3. Thus, the graph should move towards the y-value of 3 from the left side of x = 4.
04

Right-Hand Limit at x = 4

The right-hand limit \(\lim _{x \rightarrow 4^{+}} f(x)=6\)\ suggests that as x approaches 4 from values greater than 4, the graph approaches the y-value of 6. Hence, as you consider x-values slightly larger than 4, the function tends toward 6.
05

Undefined at x = 4

Since the function is not defined at \(x = 4\), there should be a noticeable break or hole at this point on the graph. Neither part of the graph from \(x < 4\) nor \(x > 4\) will connect at \(x = 4\). This discontinuity should be visible in the sketch.
06

Sketch the Graph

Using all this information, draw axes and plot points: as x approaches 2, the curve should get near a y-value of 1 without a jump. Around x = 4, draw two separate points or small lines towards y = 3 (left) and y = 6 (right) without connecting through x = 4. Indicate the small hole at x = 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Graph Sketching
When you begin sketching the graph of a function, it鈥檚 helpful to first consider the key behaviors given by limits. Limits help us understand how a function behaves as its inputs approach certain values. For our function, we have two important limit points: as \(x\) approaches 2, and as \(x\) approaches 4.

To sketch this graph:
  • Mark crucial points on the \(x\)-axis where limits are specified, such as \(x=2\) and \(x=4\).
  • For \(x=2\), draw the curve moving smoothly towards the point on the graph where \(y\) is close to 1.
  • When \(x\) is near 4, understand that different behaviors happen from each side, creating a gap at \(x=4\).
The visual representation will help exhibit how the function behaves near these important \(x\)-values.
Interpreting Discontinuity
Discontinuity in a graph indicates a break or gap where the function is not smooth or connected. In our scenario, \(x=4\) is a point of discontinuity because the function is not defined there. This means if you follow the graph from the left and right of \(x=4\), you won't find a connecting point at \(x=4\).

This is reflected as follows:
  • The function approaches a y-value of 3 from the left of \(x=4\).
  • It moves to a y-value of 6 from the right of \(x=4\).
A key visual clue to this discontinuity would be a gap or hole on the graph at \(x=4\). Such breaks remind us that the function doesn't have a continuous path through that \(x\) value.
Analyzing the Left-Hand Limit
A left-hand limit describes how a function behaves as \(x\) gets close to a specific point from the left. In our exercise, \(\lim _{x \rightarrow 4^{-}} f(x)=3\) implies that when \(x\) approaches 4 from numbers slightly less than 4, the function nears the value 3.

To depict this on a graph:
  • Imagine moving along the \(x\)-axis from smaller numbers up towards 4.
  • The graph should trend toward the y-value of 3 as it approaches \(x=4\) from the left.
This only concerns the approach from the left-hand side and doesn鈥檛 necessarily mean the function reaches 3 at 4, especially since there is a discontinuity.
Exploring the Right-Hand Limit
A right-hand limit looks at the behavior of a function as \(x\) approaches a point from the right. Here, \(\lim _{x \rightarrow 4^{+}} f(x)=6\) suggests that as \(x\) nears 4 from numbers slightly greater than 4, the function moves towards the value 6.

To visualize this limit consider:
  • Approaching 4 from the right with values greater than 4 on the \(x\)-axis.
  • The graph should slope toward the y-value of 6 without connecting at \(x=4\).
In drawings, this side would reflect how the function veers toward 6 and forms a separate point or path distinct from the left-hand side.

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