Problem 20 Use the formal definition of lim... [FREE SOLUTION]

Chapter 3: Problem 20

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 0^{-}} \frac{-4}{x}=\infty. $$

Short Answer

Expert verified

As $ x \rightarrow 0^{-} $, $ \frac{-4}{x} \rightarrow \infty $ since it grows arbitrarily large by choosing $ \delta = -\frac{4}{M} $.

Step by step solution

Understanding the Limit Definition

To prove that $ \lim _{x \rightarrow 0^{-}} \frac{-4}{x} = \infty $ using the formal definition of limits, we must show that for every positive number $ M $, there exists a $ \delta > 0 $ such that if $ 0 > x > -\delta $, then $ \frac{-4}{x} > M $. This means as $ x $ approaches zero from the left (negative side), the expression $ \frac{-4}{x} $ becomes arbitrarily large.

Setting Up the Inequality

Let's start by working with the inequality $ \frac{-4}{x} > M $. We want to find $ x $ such that this inequality holds for a given $ M > 0 $. Rearranging gives $ -4 > Mx $, or $ x > -\frac{4}{M} $. This means we must restrict $ x $ to values greater than $-\frac{4}{M}$ to satisfy the inequality.

Choosing $ \delta $

Since the limit is considered from the left side as $ x \rightarrow 0^- $, and given the inequality $ x > -\frac{4}{M} $, we can choose $ \delta = -\frac{4}{M} $. This $ \delta $ ensures that $ x $ values between zero and $-\delta$ will satisfy the condition.

Verifying the Choice of $ \delta $

Given $ x > -\delta = -(-\frac{4}{M}) = \frac{4}{M} $ and $ x < 0 $, ensuring $ x $ is slightly less than zero but greater than $-\frac{4}{M}$, we see that $ x $ does not actually need to be positive, just closer to zero from the left with a smaller absolute value than $ \frac{4}{M} $, confirming $ \frac{-4}{x} > M $.

Conclusion

Since for any $ M > 0 $, such a $ \delta$ can be chosen to make $ \frac{-4}{x} > M $ when $ x $ approaches zero from the negative side, we've shown by this definition that $ \lim _{x \rightarrow 0^{-}} \frac{-4}{x} = \infty $.

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Key Concepts

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

Formal Definition of Limits

The formal definition of a limit is foundational in understanding how a function behaves as it approaches a certain point. When we say that a limit exists for a function as the variable approaches a certain point, we're essentially stating that the function's value can get as close as desired to a certain number. Here's how it works:

We say $ \lim_{x \to a} f(x) = L $ if and only if for every number $ \epsilon > 0 $, there exists a number $ \delta > 0 $ such that if $ 0 < |x - a| < \delta $, then $ |f(x) - L| < \epsilon $.
For the limit to tend towards infinity, as in the expression $ \lim_{x \to a} f(x) = \infty $, the function needs to grow larger without bound.

This definition becomes more intuitive when applied. Basically, no matter how large a number you choose, there is an interval around the point $ a $ where the function exceeds that number. In our example, this means as $ x $ nears zero from the left side, the expression $ \frac{-4}{x} $ grows arbitrarily large, supporting an infinite limit.

Left-hand Limits

When discussing limits, looking from the left or the right can yield different behaviors in the function. Left-hand limits focus on approaching from the negative side.

Consider $ \lim_{x \to a^-} f(x) = L $, which means as $ x $ approaches $ a $ from values less than $ a $, $ f(x) $ approaches $ L $.
In a left-hand limit, you are only considering the behavior of the function from one direction, which can be crucial if a function behaves differently when approached from the other side.

For our function $ \frac{-4}{x} $, approaching zero from the left indicates the variable $ x $ is negative and moving towards zero. Thus, the negative divide makes the overall expression become positive and grow larger, leading to the limit being infinite. This examination confirms the left-sided behavior of the function.

Infinite Limits

Infinite limits occur when a function increases or decreases without bound as it approaches a certain value. It signifies that the function does not settle towards a particular finite value.

The statement $ \lim_{x \to a} f(x) = \infty $ implies that the function $ f(x) $ will surpass any arbitrarily large positive value as $ x $ closes in on $ a $.
Infinite limits are a way to describe the behavior of functions that blow up as they approach a point, rather than converging to a single number.

In our exercise, as $ x $ approaches zero from the negative side, dividing a constant by increasingly smaller negative numbers results in larger positive outputs. As a result, the limit of $ \frac{-4}{x} $ becomes infinite, indicating that no finite ceiling limits its growth. These infinite behavior scenarios are fascinating as they highlight the extreme measures a function can take when conditions change subtly.

91影视

Use the formal definition of limits to prove each statement. $$ \lim _{x \rightarrow 0^{-}} \frac{-4}{x}=\infty. $$

Short Answer

Step by step solution

Understanding the Limit Definition

Setting Up the Inequality

Choosing \( \delta \)

Verifying the Choice of \( \delta \)

Conclusion

Key Concepts

Formal Definition of Limits

Left-hand Limits

Infinite Limits

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