Question

If \(X\) remains finite and \(Y\) approaches 0 as its limit, show that $$ \lim (X Y)=0 $$

Short answer

Question: Show that as Y approaches 0, the product of a finite number X and Y also approaches 0. Answer: The result can be expressed as the limit of the product XY as Y approaches 0. Using the properties of limits, we can demonstrate that the limit of XY equals the product of the limits of X and Y individually. Since X is a finite constant, its limit is its value, while the limit of Y is 0. This shows that the limit of the product XY is 0, meaning that as Y approaches 0, the product XY also approaches 0.

Step-by-step solution

Write the given information

We are given that \(X\) remains finite and \(Y\) approaches 0 as its limit. We can express this information as follows: $$ X \in \mathbb{R} \quad \text{and} \quad \lim_{y\to0} Y = 0 $$

Analyze the product XY

Now let's consider the product of \(X\) and \(Y\). We want to find the limit for this product: $$ \lim_{y\rightarrow0}(XY) $$

Apply limit properties

We know that the limit of the product is the product of the limits, and we can apply this property to the given limit: $$ \lim_{y\rightarrow0}(XY) = \lim_{y\rightarrow0}(X) \lim_{y\rightarrow0}(Y) $$

Evaluate the limits

Since \(X\) is a finite constant, its limit is simply its value. Since \(Y\) approaches 0 as its limit, we can substitute the values into the expression: $$ \lim_{y\rightarrow0}(X) = X \quad \text{and} \quad \lim_{y\rightarrow0}(Y) = 0 $$ So our expression becomes: $$ \lim_{y\rightarrow0}(XY) = X \cdot 0 $$

Conclude the result

Finally, we know that the product of any finite number and 0 is 0. Therefore, $$ \lim_{y\rightarrow0}(XY) = 0 $$ This confirms the result that as \(Y\) approaches 0, the product \(XY\) also approaches 0.

Key concepts

Finite Values

A value is said to be finite if it is a specific, fixed number, and not approaching infinity or negative infinity. In other words, finite values are numbers that can be measured and are not extremely large or extremely small.

When discussing limits in calculus, understanding whether a term is finite is important because it affects how the limit of a product is computed. If a quantity, such as \(X\) in the given exercise, is finite, it remains constant as the limit is applied. This finite nature allows us to directly substitute \(X\) into our computations without worrying about any change when taking its limit.

In the context of the exercise, \(X\) being finite means it doesn't approach any extreme value itself as \(Y\) approaches 0. Thus, \(\lim_{y \to 0} X = X.\) This simplification significantly aids in calculating the limit of a product, which brings us to the next concept.

Limit Properties

The concept of limits is fundamental in calculus and helps us understand how functions behave near specific points. One of the essential rules involves the limits of products, sums, differences, and quotients.

For the exercise we're covering, the limit property of interest is that the limit of a product is equal to the product of the limits. Mathematically, this is expressed as:
\[ \lim_{x \to c}(f(x)g(x)) = (\lim_{x \to c} f(x))(\lim_{x \to c} g(x))\] This rule holds as long as the limits of \(f(x)\) and \(g(x)\) exist separately.

• For a constant function, such as \(f(x) = X\), the limit is straightforward: it equals the constant itself.
• In our exercise, since \(Y\) approaches 0, the relevant limit for \(Y\) is 0.

Using this property, we separate and apply the limits to each part of the product, leading to the next assumption about the product of limits.

Product of Limits

Understanding the outcome when calculating the limit of a product is critical, especially when one of the terms heads towards zero. As mentioned earlier, the limit of the product is the product of the limits, which in this scenario is expressed as:
\[\lim_{y \to 0} (XY) = (\lim_{y \to 0} X)(\lim_{y \to 0} Y)\]

Here, \(\lim_{y \to 0} X = X\) since \(X\) is finite and constant, and \(\lim_{y \to 0} Y = 0\). So, we have:
\[\lim_{y \to 0} (XY) = X \cdot 0\]
This is a clear example of how multiplying a finite number by zero always gives zero. Therefore, even if \(X\) retains any value, the entire product will always equal zero as \(Y\) approaches zero.

In limit problems like these, recognizing how the identity property of multiplication aids us in drawing straightforward conclusions is particularly helpful. This property succinctly shows why \(XY\)'s limit approaches zero in our problem.