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Chapter 15: Problem 13

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(2,9)} 101$$

Short Answer

Expert verified
Question: Evaluate the limit of the constant function f(x, y) = 101 as (x, y) approaches the point (2, 9). Answer: The limit is 101.

Step by step solution

01

Identify the function

The given function is a constant function, with the value of 101 at any point (x, y). Therefore, the function is as follows: f(x, y) = 101.
02

Evaluate the limit

Since the function is a constant, its value does not change as (x, y) approaches any point. Therefore, the limit when (x, y) approaches (2, 9) is simply the value of the function, that is: $$\lim _{(x, y) \rightarrow(2,9)} 101 = 101$$

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Key Concepts

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

Constant Function
In calculus, a constant function is one where the output value remains the same for any input value. This means the function does not change and is unaffected by variations in the input variables.
For example, the function given in the exercise, \( f(x, y) = 101 \), remains constant regardless of the values of \( x \) and \( y \).
This is because the output is a fixed number - in this case, 101 - thus it's unswayed by the transition of input variables.
This simplicity is why constant functions often play a pivotal role when evaluating limits, as the constant value directly equals the limit.
Multivariable Limits
Multivariable limits extend the concept of limits from single-variable calculus to functions of multiple variables, like \( f(x, y) \).
They describe how the function behaves as its input approaches a particular point in a multidimensional space.
When evaluating multivariable limits, it's essential to understand that the variables may approach from any direction within the space.
However, with a constant function, no matter how the variables, \( (x, y) \) in this case, approach a point, the limit remains the fixed value of the function.
This characteristic simplifies calculations significantly compared to non-constant multivariable functions.
Evaluating Limits
When evaluating limits, especially in multivariable calculus, you assess how the output of a function behaves as the inputs get close to a specified point.
For constant functions, this process is straightforward since the output's value never changes.
Thus, in evaluating a limit like \( \lim _{(x, y) \rightarrow(2,9)} 101 \), it directly results in the constant value itself, 101.
With non-constant functions, more intricate methods are often necessary, including analyzing the function's behavior from multiple directions.
  • Constant functions: Simplify limit evaluation by providing immediate solutions.
  • Non-constant functions: May require different approaches depending on the function's complexity.
Understanding how to evaluate these limits is crucial for mastering multivariable calculus skills effectively.

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Most popular questions from this chapter

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