Question

Tell me the limit class 11 formula

Answer 1

In Class 11 mathematics, limits are an essential concept in calculus. Here are some important limit formulas that you may encounter:

1. **Basic Limit Formulas:**
\[
\lim_{{x \to a}} c = c \quad \text{(where } c \text{ is a constant)}
\]
\[
\lim_{{x \to a}} x = a
\]
\[
\lim_{{x \to a}} (x + b) = a + b \quad \text{(where } b \text{ is a constant)}
\]

2. **Limit of a Sum:**
\[
\lim_{{x \to a}} (f(x) + g(x)) = \lim_{{x \to a}} f(x) + \lim_{{x \to a}} g(x)
\]

3. **Limit of a Difference:**
\[
\lim_{{x \to a}} (f(x) - g(x)) = \lim_{{x \to a}} f(x) - \lim_{{x \to a}} g(x)
\]

4. **Limit of a Product:**
\[
\lim_{{x \to a}} (f(x) \cdot g(x)) = \lim_{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x)
\]

5. **Limit of a Quotient:**
\[
\lim_{{x \to a}} \left( \frac{f(x)}{g(x)} \right) = \frac{\lim_{{x \to a}} f(x)}{\lim_{{x \to a}} g(x)} \quad \text{(provided } \lim_{{x \to a}} g(x) \neq 0\text{)}
\]

6. **Limit of a Constant Multiple:**
\[
\lim_{{x \to a}} (c \cdot f(x)) = c \cdot \lim_{{x \to a}} f(x) \quad \text{(where } c \text{ is a constant)}
\]

7. **Limit of \(x^n\) as \(x\) approaches \(a\):**
\[
\lim_{{x \to a}} x^n = a^n \quad \text{(for any real number } n\text{)}
\]

8. **Squeeze Theorem:**
If \(f(x) \leq g(x) \leq h(x)\) for all \(x\) in some interval around \(a\) (except possibly at \(a\)), and
\[
\lim_{{x \to a}} f(x) = \lim_{{x \to a}} h(x) = L,
\]
then
\[
\lim_{{x \to a}} g(x) = L.
\]

9. **Special Limits:**
\[
\lim_{{x \to 0}} \frac{\sin x}{x} = 1
\]
\[
\lim_{{x \to 0}} \frac{1 - \cos x}{x^2} = \frac{1}{2}
\]
\[
\lim_{{x \to \infty}} \frac{1}{x} = 0
\]
\[
\lim_{{x \to \infty}} \frac{1}{x^n} = 0 \quad \text{(for any positive integer } n\text{)}
\]

These formulas are fundamental to solving limits and are often used in conjunction with limit properties.