Limit formula

There isn’t a single formula that covers all types of limits, as limits can take various forms depending on the function and the point at which you’re evaluating the limit. However, there are some fundamental limit formulas and rules that are commonly used in calculus. Here are some of them:

  1. Direct Substitution: If you can directly substitute the value that x is approaching into the function without encountering an issue, you do so. For example:

    lim (x → a) f(x) = f(a) if f(a) is defined.

  2. Limits of Constants: The limit of a constant is just the constant itself:

    lim (x → a) c = c, where c is a constant.

  3. Limits of Linear Functions: The limit of a linear function is straightforward:

    lim (x → a) (mx + b) = ma + b, where m and b are constants.

  4. Limits of Powers of x: For simple powers of x, you have:

    a. lim (x → a) x^n = a^n, where n is a constant. b. lim (x → ∞) x^n = ∞ if n > 0 and 0 if 0 < n < 1. c. lim (x → 0) x^n = 0 if n > 0.

  5. Limits of Rational Functions: For rational functions (quotients of polynomials), you can often use direct substitution. If the denominator is zero at the limit point, it may be an indeterminate form (0/0), requiring further analysis.
  6. Limits of Trigonometric Functions:

    a. lim (x → 0) sin(x)/x = 1. b. lim (x → 0) (1 – cos(x))/x = 0.

  7. Limits of Exponential and Logarithmic Functions:

    a. lim (x → ∞) e^x = ∞. b. lim (x → 0) ln(x) = -∞.

  8. Limits at Infinity:

    a. lim (x → ∞) f(x) = L if the function approaches a finite limit L as x goes to infinity. b. lim (x → -∞) f(x) = L if the function approaches a finite limit L as x goes to negative infinity.

  9. Limits of Sums and Products:

    a. lim (x → a) [f(x) + g(x)] = lim (x → a) f(x) + lim (x → a) g(x). b. lim (x → a) [f(x) * g(x)] = [lim (x → a) f(x)] * [lim (x → a) g(x)].

  10. Limits of Compositions: If f(x) and g(x) are functions, you can sometimes find the limit of their composition as follows:

    lim (x → a) [f(g(x))] = f[lim (x → a) g(x)], if the limit on the right exists.

  11. Squeeze Theorem: If you have three functions f(x), g(x), and h(x) such that f(x) ≤ g(x) ≤ h(x) for all x in some interval around ‘a,’ and lim (x → a) f(x) = lim (x → a) h(x) = L, then lim (x → a) g(x) = L.
  12. L’Hôpital’s Rule: When you have an indeterminate form (0/0 or ∞/∞), you can apply L’Hôpital’s Rule, which involves taking the derivatives of the numerator and denominator repeatedly.

These are some of the fundamental limit rules and formulas used in calculus. The specific rule you apply depends on the form of the limit and the function you’re working with. It’s essential to understand when and how to use these rules to evaluate limits effectively.

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