Problem 26 Find each limit, if it exists: ... [FREE SOLUTION]

Chapter 2: Problem 26

Find each limit, if it exists: |a) $\lim _{x \rightarrow a^{-}} f(x)$ (b) $\lim _{x \rightarrow a^{+}} f(x)$ (c) $\lim _{x \rightarrow a} f(x)$ $$ f(x)=\frac{x+5}{|x+5|}: \quad a=-5 $$

Short Answer

Expert verified

a) -1, b) 1, c) does not exist.

Step by step solution

Understand the Function

The given function is $ f(x) = \frac{x+5}{|x+5|} $. The absolute value $|x+5|$ affects whether the function will use $x+5$ or $-(x+5)$ depending on the value of $x$ in relation to $a = -5$.

Consider Left-Hand Limit

First, calculate the left-hand limit $\lim_{x \to a^{-}} f(x)$. For $x$ approaching $-5$ from the left, $x+5 < 0$, so $|x+5| = -(x+5)$. Thus, the function becomes $-1$. The left-hand limit is $\lim_{x \to -5^{-}} f(x) = -1$.

Consider Right-Hand Limit

Next, calculate the right-hand limit $\lim_{x \to a^{+}} f(x)$. For $x$ approaching $-5$ from the right, $x+5 > 0$, so $|x+5| = x+5$. Thus, the function becomes $1$. The right-hand limit is $\lim_{x \to -5^{+}} f(x) = 1$.

Evaluate Two-Sided Limit

For the two-sided limit $\lim_{x \to a} f(x)$, both the left and right-hand limits must be equal. Here, they are not equal (left-hand limit is $-1$ and right-hand limit is $1$). Therefore, $\lim_{x \to -5} f(x)$ does not exist because the function has a jump discontinuity at $x = -5$.

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Key Concepts

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

Understanding Left-Hand Limit

When we talk about the left-hand limit, we mean the behavior of a function as the input, or $ x $, approaches a specific value from the left side. Picture a number line. For the left-hand limit, you're moving towards the number from smaller values.
In our exercise, this means we're examining what happens to the function $ f(x) = \frac{x+5}{|x+5|} $ as $ x $ approaches $ -5 $ from values less than $ -5 $.

For $ x < -5 $, $ x + 5 $ is negative.
The absolute value $ |x+5| $ turns $ x+5 $ into $ -(x+5) $.
This changes the function to $ \frac{x+5}{-(x+5)} = -1 $.

Thus, the left-hand limit, $ \lim _{x \to -5^-} f(x) $, equals $-1$. This shows how critical it is to understand each side of the value to see how the function behaves.

Exploring Right-Hand Limit

The right-hand limit checks the function's behavior as $ x $ approaches a certain value from numbers larger than that specific point.
In this particular exercise, we're looking at values of $ x $ greater than $ -5 $, so approaching from the right side.

For $ x > -5 $, $ x + 5 $ becomes positive.
The absolute value $ |x+5| $ remains the same, $ x+5 $.
The function then simplifies to $ \frac{x+5}{x+5} = 1 $.

Therefore, the right-hand limit $ \lim _{x \to -5^+} f(x) $ is $ 1 $.
This process underscores the importance of understanding how different approaches affect the function's outcome.

Recognizing Two-Sided Limit

The two-sided limit is all about consistency. It requires the left and right-hand limits at a point to be identical. If they aren't, the two-sided limit doesn't exist.
For our function, as $ x $ approaches $ -5 $:

Left-hand limit is $-1$.
Right-hand limit is $1$.

Since $-1$ and $1$ are not equal, the function has what's known as a jump discontinuity at $ x = -5 $.
This causes the two-sided limit $ \lim _{x \to -5} f(x) $ to not exist. That's an essential point: if there's a jump, the function has no consistent value from both sides.

Role of Absolute Value Function

The absolute value function $ |x+5| $ plays a pivotal role in determining how $ f(x) $ behaves near the point of interest. Normally, the absolute value function takes any real number and makes it non-negative. This behavior influences the nature of the limits.
Consider:

For $ x < -5 $, where $ x+5 $ is negative, $ |x+5| = -(x+5) $, providing a negative one-sided output.
For $ x > -5 $, $ x+5 $ is positive, and thus $ |x+5| = x+5 $.

Understanding the properties of absolute value is key to analyzing limits and continuity.
In mathematical expressions, it defines the major differences in the behavior of functions like jump discontinuities, as observed in our example.

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91影视

Find each limit, if it exists: |a) \(\lim _{x \rightarrow a^{-}} f(x)\) (b) \(\lim _{x \rightarrow a^{+}} f(x)\) (c) \(\lim _{x \rightarrow a} f(x)\) $$ f(x)=\frac{x+5}{|x+5|}: \quad a=-5 $$

Short Answer

Step by step solution

Understand the Function

Consider Left-Hand Limit

Consider Right-Hand Limit

Evaluate Two-Sided Limit

Key Concepts

Understanding Left-Hand Limit

Exploring Right-Hand Limit

Recognizing Two-Sided Limit

Role of Absolute Value Function

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