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Limits help us study the behavior of a function as it approaches a particular point from either side. Here, we are interested in how the function behaves as it approaches 0 from the left and the right.
For the function given, as the variable approaches 0 from the left (noted as \( x \to 0^- \)), it is defined by \( f(x) = \cos x \). The limit from the left is \( \cos 0 = 1 \).
Conversely, as the variable approaches 0 from the right (denoted as \( x \to 0^+ \)), the function is \( f(x) = 1 - x^2 \). In this case, the limit from the right is also 1 because \( 1 - 0^2 = 1 \).
This concept is essential for identifying discontinuities in functions where limits from both sides and the actual function value at a point may differ.

The function's behavior changes based on different input ranges, which is typical for piecewise functions. Here we have a piecewise rule.

• For \( x < 0 \): The function behaves like \( f(x) = \cos x \), oscillating between -1 and 1.
• For \( x = 0 \): The function has a constant value \( f(x) = 0 \).
• For \( x > 0 \): It behaves according to \( f(x) = 1-x^2 \), forming a downward-facing parabola starting from \( 1 \).

Observing such function behavior helps in understanding which part of the function defines the output for different intervals.

Graph sketching is about transforming function rules into visual representations. It helps us understand function behavior and discontinuities.
To sketch the function described:

• For \( x < 0 \), plot the cosine curve which smoothly oscillates. It's crucial to notice it approaches \( y = 1 \) as \( x \) nears 0 from the left.
• Mark a distinct point at \( (0,0) \) since \( f(x) \) is explicitly 0 at \( x = 0 \).
• For \( x > 0 \), draw a parabola starting at 1 with vertex at \( x = 0 \). This visual representation highlights the discontinuity at \( x = 0 \).

Sketching these segments will visually illustrate the behavior and jump in the function values at this key point.

The left-hand limit checks the function's behavior as we approach a point from the left side.
For this function, as we approach \( x = 0 \) from the left, we use \( f(x) = \cos x \). Hence, the left-hand limit is \( \cos 0 = 1 \).
Even though the limit approaches 1, the function value right at \( x = 0 \) is 0, which indicates a discontinuity.
Understanding this difference in behavior between the limit and function value is crucial for identifying where functions break or jump at specific points.

The right-hand limit considers how a function behaves as we near a given point from the right.
For values approaching 0 from the right in this function, we use \( f(x) = 1 - x^2 \). The calculation shows the right-hand limit is \( 1 - 0^2 = 1 \).
This right-hand limit coincides with the left-hand limit, both being 1. Yet, neither matches the actual function value at \( x = 0 \), which is 0.
This indicates a jump discontinuity at \( x = 0 \), an interruption seen in the graph, aligning with shifts in expected behavior.