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Limit notation is a mathematical way of expressing the behavior of a function as its input values approach a specific point, often infinity or negative infinity. For exponential functions like \( y(x) = 1.5^x \), we are interested in what happens as \( x \) becomes very large or very small.

In our exercise, we described the behavior for very large \( x \) (\( x \to +\infty \)) and very small \( x \) (\( x \to -\infty \)). This behavior can be precisely communicated using limit notation. Here, it reads:

• As \( x \) approaches positive infinity, the function \( 1.5^x \) increases towards infinity. In limit notation, this is \( \lim_{{x \to +\infty}} 1.5^x = +\infty \).
• As \( x \) approaches negative infinity, the function \( 1.5^x \) decreases towards zero. In limit notation, this is \( \lim_{{x \to -\infty}} 1.5^x = 0 \).

Think of limit notation as a shorthand for describing trends in a function's output values, helping to quickly comprehend the long-term behavior of the function.

End behavior describes how a function behaves as its input approaches extreme values, either very large (positive infinity) or very small (negative infinity). In the context of exponential functions, understanding end behavior helps us predict how the graph will evolve.

With the function \( y(x) = 1.5^x \), we identified that as \( x \to +\infty \), the output \( y \) tends to grow without bound. This tells us that when we look far right on the graph, the curve is shooting upwards. Similarly, as \( x \to -\infty \), the output \( y \) gets closer and closer to zero. This implies that far to the left on the graph, the line approaches, but never quite touches, the x-axis.

In essence, end behavior draws a picture of the function's graph primarily focusing on what happens at the ends of the x-values.

• For \( x \to +\infty \), the curve rises sharply.
• For \( x \to -\infty \), the curve flattens out approaching zero.

Understanding end behavior is crucial for predicting function behavior without needing to compute numerous points.

A horizontal asymptote is a horizontal line that a graph approaches as \( x \) goes to positive or negative infinity. It gives a boundary for the value of the function but doesn't get completely reached in most cases. Exponential functions, like many others, have horizontal asymptotes that guide this understanding.

For the expression \( y(x) = 1.5^x \), the concept of a horizontal asymptote is paramount as \( x \to -\infty \), since this is where the graph's path becomes predictable and smooths out. The function moves closer and closer to the line \( y = 0 \), but never actually meets it.

• The horizontal asymptote for \( y(x) = 1.5^x \) is \( y = 0 \)

Why is this important? The horizontal asymptote helps in understanding the 'flattening effect' seen in the graph on the left end and signals that however much smaller \( x \) gets, the function never goes below zero.

Remember: horizontal asymptotes act like invisible guides showing how the graph behaves when extended in both directions to infinity.