The concept of "limit change" can refer to several different ideas depending on the context, such as changing the form of a limit, converting a limit into another form, or transforming variables in a limit. Below are a few key formulas and concepts related to limits and transformations in calculus:

1. **Change of Variable in Limits (Substitution):**
   If \( \lim_{x \to c} f(x) \) exists and \( x = g(t) \) where \( g \) is a continuous function such that \( g(t) \to c \) as \( t \to t_0 \), then:
   \[
   \lim_{x \to c} f(x) = \lim_{t \to t_0} f(g(t))
   \]

2. **L'Hôpital's Rule:**
   This rule is used to evaluate limits of indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). If \( \lim_{x \to c} \frac{f(x)}{g(x)} \) yields an indeterminate form, then:
   \[
   \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}
   \]
   provided the limit on the right exists.

3. **Squeeze Theorem:**
   If \( f(x) \leq g(x) \leq h(x) \) for all \( x \) near \( c \) (except possibly at \( c \)) and \( \lim_{x \to c} f(x) = \lim_{x \to c} h(x) = L \), then:
   \[
   \lim_{x \to c} g(x) = L
   \]

4. **Limit of a Composite Function:**
   If \( \lim_{x \to c} g(x) = L \) and \( \lim_{y \to L} f(y) \) exists, then:
   \[
   \lim_{x \to c} f(g(x)) = \lim_{y \to L} f(y)
   \]

5. **Infinite Limits and Asymptotes:**
   For functions that approach infinity, we define:
   \[
   \lim_{x \to \infty} f(x) = L \quad \text{(if \( f(x) \to L \) as \( x \to \infty \))}
   \]
   \[
   \lim_{x \to c} f(x) = \infty \quad \text{(if \( f(x) \to \infty \) as \( x \to c \))}
   \]

Would you like further details or examples on any of these concepts?

Here are 5 questions you might consider asking next:
1. How do you apply L'Hôpital's Rule in different indeterminate forms?
2. What is the Squeeze Theorem and how is it used in solving limits?
3. How can substitution be used to simplify limit problems?
4. What are the common techniques for finding limits involving infinity?
5. Can you explain the concept of continuity and its relation to limits?

**Tip:** When solving limit problems, it's crucial to first identify the type of limit you're dealing with (e.g., finite, infinite, indeterminate form) to choose the most appropriate technique.

Explore key formulas and theorems related to limit changes in calculus, including Change of Variable, L'Hôpital's Rule, Squeeze Theorem, and more. Perfect for advanced students and learners interested in calculus transformations.

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