find a two functions f(x) and g(x) where the limit of f(x) as x approaches zero and the limit of g(x) as x approaches zero don’t exist but the limit of f(x)/g(x) as x approaches zero does exist

To find two functions \( f(x) \) and \( g(x) \) such that:

1. \( \lim_{x \to 0} f(x) \) does not exist,
2. \( \lim_{x \to 0} g(x) \) does not exist,
3. \( \lim_{x \to 0} \frac{f(x)}{g(x)} \) exists,

we can consider oscillatory functions that have undefined limits at \( x = 0 \), but their ratio produces a well-defined limit.

One good pair of functions that meets these criteria is:

- \( f(x) = \sin\left(\frac{1}{x}\right) \)
- \( g(x) = \frac{1}{x} \sin\left(\frac{1}{x}\right) \)

### Why these functions work:

1. **Limit of \( f(x) = \sin\left(\frac{1}{x}\right) \) as \( x \to 0 \):**
   - This function oscillates between -1 and 1 as \( x \to 0 \), so the limit does not exist.
   
2. **Limit of \( g(x) = \frac{1}{x} \sin\left(\frac{1}{x}\right) \) as \( x \to 0 \):**
   - This also oscillates, and grows unbounded as \( x \to 0 \), so the limit does not exist.

3. **Limit of \( \frac{f(x)}{g(x)} \) as \( x \to 0 \):**
   \[
   \frac{f(x)}{g(x)} = \frac{\sin\left(\frac{1}{x}\right)}{\frac{1}{x} \sin\left(\frac{1}{x}\right)} = x
   \]
   - The limit of \( x \) as \( x \to 0 \) is clearly 0, so the limit of the ratio exists and is 0.

Thus, we have found two functions where the individual limits do not exist, but their ratio has a well-defined limit of 0 as \( x \to 0 \).

Would you like further explanation on this example? Here are 5 related questions:

1. Can we choose other pairs of oscillatory functions for similar results?
2. What happens if we use different scaling factors for \( f(x) \) and \( g(x) \)?
3. How can we prove that these limits do not exist rigorously?
4. Could we use piecewise-defined functions to achieve similar behavior?
5. How would these limits change if we approached a different point other than zero?

**Tip:** When solving limit problems, particularly with undefined behaviors, try looking for patterns in oscillatory functions or division by zero scenarios to explore more interesting cases.

Find two functions f(x) and g(x) where the limit of f(x) as x approaches zero and the limit of g(x) as x approaches zero don’t exist, but the limit of f(x)/g(x) as x approaches zero does exist.

Explore how to find two functions f(x) = sin(1/x) and g(x) = (1/x) * sin(1/x) where individual limits as x approaches zero do not exist, but the limit of their ratio exists and is well-defined. Learn key mathematical concepts like oscillatory functions and limits.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation