Problem 21 - 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. \(f(x)=\left\\{\begin{array}{ll}x & \text { if } x<1 \\ 2 & \text { if } x=1 ; \quad a=1 \\ -x+2 & \text { if } x>1\end{array}\right.\)

Step-by-Step Solution

Answer

In summary, for the given function \(f(x)\) and given value \(a=1\), we have: a) \(\lim_{x \rightarrow 1^{-}} f(x) = 1\) b) \(\lim_{x \rightarrow 1^{+}} f(x) = 1\) c) \(\lim_{x \rightarrow 1} f(x) = 1\)

Step 1: Sketch the graph of the function \(f(x)\)

We have a piecewise function consisting of three cases: - For \(x<1\), \(f(x) = x\) - For \(x=1\), \(f(x) = 2\) - For \(x>1\), \(f(x) = -x + 2\) To sketch the graph of this function, we'll draw each of these segments in their respective domains while considering the given value of \(a=1\).

Step 2: Evaluate \(\lim _{x \rightarrow a^{-}} f(x)\) for \(a=1\)

To evaluate the left-hand limit of the function at \(a=1\), we'll consider the case when \(x<1\). In this region, \(f(x) = x\). As \(x\) approaches \(1\) from the left, the function behaves as a line with a slope of \(1\): \[\lim_{x \rightarrow 1^{-}} f(x) = \lim_{x \rightarrow 1^{-}} x = 1.\]

Step 3: Evaluate \(\lim _{x \rightarrow a^{+}} f(x)\) for \(a=1\)

Now, we'll evaluate the right-hand limit of the function at \(a=1\). In the case when \(x>1\), we have \(f(x)=-x+2\). As \(x\) gets closer to \(1\) from the right, the function behaves as a line with a slope of \(-1\): \[\lim_{x \rightarrow 1^{+}} f(x) = \lim_{x \rightarrow 1^{+}} (-x + 2) = 1.\]

Step 4: Evaluate \(\lim _{x \rightarrow a} f(x)\) for \(a=1\)

Finally, we can evaluate the limit of \(f(x)\) at \(a=1\). Since both the left-hand limit and the right-hand limit are equal and finite, the limit of the function exists at \(a = 1\): \[\lim_{x \rightarrow 1} f(x) = \lim_{x \rightarrow 1^{-}} f(x) = \lim_{x \rightarrow 1^{+}} f(x) = 1.\] In summary, we have found the following limits for \(f(x)\) at \(a=1\): a) \(\lim_{x \rightarrow 1^{-}} f(x) = 1\) b) \(\lim_{x \rightarrow 1^{+}} f(x) = 1\) c) \(\lim_{x \rightarrow 1} f(x) = 1\)

Key Concepts

Limits

Limits are a fundamental concept in calculus that help us understand the behavior of functions as the input (x-value) approaches a certain point. In our exercise, we are interested in the limit behavior of a piecewise function at a particular point, `a = 1`.

When assessing limits, we want to consider what happens to the value of the function as we get infinitely close to a specific point from both directions. This means looking at the values from slightly less than and slightly more than the point within the function's domain.

• If the function approaches the same value from both sides, the limit exists and is equal to that value.
• If the function approaches different values from either side, the limit does not exist at that point.

In our piecewise function exercise, we evaluate separate limits for when `x` is approaching the point from the left and from the right, and then assess the overall limit at `a = 1`.

Graph Sketching

Graph sketching is the art of drawing a rough image of a function to understand its general shape and behavior. For piecewise functions like the one in our exercise, graph sketching involves plotting each piece in their respective domains.

To sketch our given function, we follow these steps:

• For `x < 1`, sketch the line `f(x) = x`, which is a diagonal line passing through the origin with a slope of `1` and continues until slightly less than `x = 1`.
• At `x = 1`, plot the point `(1, 2)` as a distinct value since `f(x) = 2` at this specific location.
• For `x > 1`, sketch the line `f(x) = -x + 2`, a downward sloping line that intersects the y-axis at `2` and extends rightward.

These pieces connect to form the entire graph of the function. Sketching helps visualize the overall behavior of the function and analyze it better, particularly near the critical point `a = 1`.

Continuity

Continuity of a function at a point means that there is no interruption in the graph at that point; the function does not have breaks, gaps, or jumps.

To determine if a function is continuous at a certain point, you need to check for three conditions:

• The function must be defined at the point.
• The limit from the left and the limit from the right as `x` approaches the point must exist and be equal.
• The limit at that point must equal the function's value at that point.

For our piecewise function at `a = 1`, although the left and right hand limits are both `1`, the function at `x = 1` gives us `2`. Therefore, there is a discontinuity at `x = 1` because the limit of the function as `x` approaches `1` does not equal the function's value at `1`, despite the one-sided limits being equal.

Left-hand Limit

The left-hand limit refers to the behavior of a function as the input approaches a particular point from the left side. In mathematical usage, this is expressed as ` $$\lim_{x \to a^-} f(x)$$,` where `a` is the point we are examining.

For our problem, we're checking the left-hand limit as `x` approaches `1` from values less than `1`. Since for `x < 1`, `f(x)` is defined by `x`, the limit aligns with the behavior of the function under this part:

\[\lim_{x \rightarrow 1^{-}} f(x) = \lim_{x \rightarrow 1^{-}} x = 1.\]
This means that the function's output approaches `1` as we approach the point from the left side.

Right-hand Limit

The right-hand limit looks at how a function behaves as the input approaches from the opposite direction or the right side. It is represented mathematically as ` $$\lim_{x \to a^+} f(x)$$,` where `a` is the specific target point.

For our piecewise function, we need to find the right-hand limit as `x` approaches `1` from values greater than `1`. For `x > 1`, `f(x)` is defined by `-x + 2`. Thus we compute the limit as follows:

\[\lim_{x \rightarrow 1^{+}} f(x) = \lim_{x \rightarrow 1^{+}} (-x + 2) = 1.\]
This shows that approaching `1` from the right also leads the function to approach the value `1`, consistent with the left-hand limit, though not the function's value at `x = 1` itself, which is `2`.