Problem 5 For the given $f(x),$ express ... [FREE SOLUTION]

Chapter 2: Problem 5

For the given $f(x),$ express each of the following limits as $\infty,-\infty,$ or DNE (Does Not Exist): (a) $\lim _{x \rightarrow a} f(x)$ (b) $\lim _{x \rightarrow a^{-}} f(x)$ (c) $\lim _{x \rightarrow a} f(x)$ $$ f(x)=\frac{3 x}{(x+8)^{2}}: \quad a=-8 $$

Short Answer

Expert verified

For all parts (a), (b), and (c), the limit is $-\infty$.

Step by step solution

Identify the function behavior around x = a

We are given the function $ f(x) = \frac{3x}{(x+8)^2} $ and we need to analyze the behavior of this function as $ x $ approaches $ a = -8 $. Notice that at $ x = -8 $, the denominator becomes zero, which suggests that the function might have a vertical asymptote at this point.

Analyze the limit from the left, $ \lim_{x \to a^-} f(x) $

Consider the limit $ \lim_{x \to -8^-} \frac{3x}{(x+8)^2} $. As $ x $ approaches $-8$ from the left, $(x+8) \to 0^-$ (a small negative number), making $ (x+8)^2 \to 0^+ $. Simultaneously, $ 3x \to -24 $. Hence, the fraction $ \frac{3x}{(x+8)^2} \to -\infty $.

Analyze the limit from the right, $ \lim_{x \to a^+} f(x) $

Consider the limit $ \lim_{x \to -8^+} \frac{3x}{(x+8)^2} $. As $ x $ approaches $-8$ from the right, $ (x+8) \to 0^+ $ and similarly $ (x+8)^2 \to 0^+ $. Meanwhile, $ 3x \to -24 $. Therefore, the fraction $ \frac{3x}{(x+8)^2} \to -\infty $.

Analyze the twosided limit, $ \lim_{x \to a} f(x) $

For the two-sided limit, $ \lim_{x \to -8} f(x) $, since both the left-hand and right-hand limits equal $-\infty$, the two-sided limit is also $-\infty$. This is because both $ \lim_{x \to -8^-} f(x) $ and $ \lim_{x \to -8^+} f(x) $ result in $-\infty$.

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Key Concepts

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

Understanding Vertical Asymptotes

A vertical asymptote is a line that a graph approaches but never touches or crosses.
For a function like our given function, it might occur when the denominator of a fraction gets very close to zero, causing the function's value to surge rapidly towards infinity or negative infinity.

In the case of the function $ f(x) = \frac{3x}{(x+8)^2} $, the denominator $(x+8)^2$ becomes zero as $x$ approaches $-8$. This leads to the potential for a vertical asymptote at $x = -8$.
Vertical asymptotes can cause distinct behavior on either side of the asymptote, resulting in important considerations when evaluating limits, especially for the left-hand and right-hand limits.

What is a Left-Hand Limit?

The left-hand limit of a function examines its behavior as the variable, like $x$, approaches a particular value from the left side. This provides insights into how the function behaves just before reaching that value.

For $ f(x) = \frac{3x}{(x+8)^2} $, we calculate the left-hand limit as $ x \to -8^- $. As the expression $(x+8)$ approaches zero from the negative side, meaning it is slightly less than zero, it squares to a positive number that is quite small. Meanwhile, $3x$ becomes a negative number which tends towards $-24$.

Combining these tendencies in the fraction $ \frac{3x}{(x+8)^2}$ results in the function moving towards $-\infty$. This observation tells us how the function behaves as it gets closer to the vertical asymptote from the left side.

Exploring Right-Hand Limit

The right-hand limit is concerned with how the function behaves as one approaches a given value from the right. This perspective helps in understanding the function's behavior just after reaching the value in question.

For the function $ f(x) = \frac{3x}{(x+8)^2} $, we assess the right-hand limit as $ x \to -8^+ $. Here, $(x+8)$ approaches zero from the positive side, leading it to be slightly greater than zero, but still positive. As $3x$ continues to head towards $-24$, similar dynamics occur as with the left-hand limit. So, the fraction $ \frac{3x}{(x+8)^2} $ directs itself towards $-\infty$ as well.
This outcome indicates that even from the right, the function's values plummet toward negative infinity as it nears the vertical asymptote at $-8$. The results of both the left-hand and right-hand limits being equal also confirm the feature of the vertical asymptote at that particular $x$-value.

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91影视

For the given \(f(x),\) express each of the following limits as \(\infty,-\infty,\) or DNE (Does Not Exist): (a) \(\lim _{x \rightarrow a} f(x)\) (b) \(\lim _{x \rightarrow a^{-}} f(x)\) (c) \(\lim _{x \rightarrow a} f(x)\) $$ f(x)=\frac{3 x}{(x+8)^{2}}: \quad a=-8 $$

Short Answer

Step by step solution

Identify the function behavior around x = a

Analyze the limit from the left, \( \lim_{x \to a^-} f(x) \)

Analyze the limit from the right, \( \lim_{x \to a^+} f(x) \)

Analyze the twosided limit, \( \lim_{x \to a} f(x) \)

Key Concepts

Understanding Vertical Asymptotes

What is a Left-Hand Limit?

Exploring Right-Hand Limit

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