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Magazine Chapter 2: Problem 49
Find each limit, if it exists: (a) \(\lim _{x \rightarrow a^{-}} f(x)\) (b) \(\lim _{x \rightarrow a^{+}} f(x)\) (c) \(\lim _{x \rightarrow a} f(x)\) $$ f(x)=\sqrt{5-x} ; \quad a=5 $$
Short Answer
Expert verified
(a) 0, (b) does not exist, (c) does not exist.
Step by step solution
01
Understanding the Function Behavior
We are given the function \(f(x)=\sqrt{5-x}\) and need to consider its behavior at \(x = 5\). The function is defined for values \(x \leq 5\) as the square root of negative numbers is not defined in real numbers.
02
Calculating Left-Hand Limit
To find \( \lim _{x \rightarrow 5^{-}} f(x) \), consider values of \(x\) that approach 5 from the left (\(x < 5\)). As \(x\) approaches 5 from the left, \(5-x\) approaches 0 and \(\sqrt{5-x}\) approaches \(0\). Thus, \( \lim _{x \rightarrow 5^{-}} f(x) = 0\).
03
Investigating Right-Hand Limit
For \( \lim _{x \rightarrow 5^{+}} f(x) \), we need values of \(x\) approaching 5 from the right (\(x > 5\)). Since the function \(f(x)=\sqrt{5-x}\) is not defined for \(x > 5\), the right-hand limit \( \lim _{x \rightarrow 5^{+}} f(x) \) does not exist.
04
Determining Two-Sided Limit
The two-sided limit \( \lim _{x \rightarrow 5} f(x) \) exists only if both the left-hand and right-hand limits exist and are equal. Here, since the right-hand limit does not exist, \( \lim _{x \rightarrow 5} f(x) \) does not exist.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Left-Hand Limit
When we talk about the left-hand limit of a function, we are looking at the values the function approaches as our variable gets closer to a specific point from the left side. In this case, as we approach the value 5 from the left for the function \(f(x)=\sqrt{5-x}\), we consider values just smaller than 5.
- For \(x < 5\), \(5-x\) becomes a small positive number, since 5 minus a number slightly less than 5 is slightly more than 0.
- As a result, \(\sqrt{5-x}\) will approach 0 as \(x\) gets closer to 5 from the left.
Right-Hand Limit
The right-hand limit involves approaching the specific point from the right side, considering values larger than the point of interest. For our function \(f(x)=\sqrt{5-x}\), this means examining values larger than 5. However, here is the catch:
- For values of \(x > 5\), \(5-x\) becomes negative.
- Since the square root of a negative number doesn't exist in the real number system, \(f(x)\) isn't defined for these values.
Two-Sided Limit
A two-sided limit requires the function to approach the same value from both the left and right sides as \(x\) approaches a specific point. Here's how it works for our example:
- We checked the left-hand limit and found it to be 0.
- However, the right-hand limit does not exist because the function isn't defined for \(x > 5\).
Undefined Functions
An undefined function is one that doesn't produce a real number output for certain inputs. In the context of our problem, the function \(f(x)=\sqrt{5-x}\) becomes undefined for values where \(5-x\) is negative, specifically for \(x > 5\).
- What this means is that the function doesn't work for inputs that result in trying to take the square root of a negative number.
- This affects the ability to compute limits in the direction where the function is undefined, impacting both the right-hand and two-sided limits.
Function Behavior
Understanding the behavior of a function is all about knowing how it changes as the input changes, especially near points of interest. For \(f(x)=\sqrt{5-x}\), the function behavior at \(x=5\) is crucial:
- For \(x \leq 5\), the function provides real values. As \(x\) approaches 5 from the left, the values approach 0.
- For \(x > 5\), the function is undefined, highlighting a boundary of function behavior.
- This leads to the conclusion that as we closely approach 5, the function's output hits a solid wall at 0, but cannot cross over for \(x > 5\).
