List of basic formulas and rules that can help in calculating limits:

  1. Limit of a constant:
    [
    \lim_{x \to a} c = c
    ]
    (where (c) is a constant)
  2. Sum limit:
    [
    \lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)
    ]
  3. Difference limit:
    [
    \lim_{x \to a} (f(x) – g(x)) = \lim_{x \to a} f(x) – \lim_{x \to a} g(x)
    ]
  4. Product limit:
    [
    \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 the quotient:
    [
    \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad g(a) \neq 0
    ]
  6. Degree limit:
    [
    \lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n
    ]
  7. L’Hôpital’s rule (for uncertainties of the form ( \frac{0}{0} ) or ( \frac{\infty}{\infty} )):
    [
    \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}
    ]
  8. Limits of trigonometric functions:
    [
    \lim_{x \to 0} \frac{\sin x}{x} = 1
    ]
    [
    \lim_{x \to 0} \frac{1 – \cos x}{x^2} = \frac{1}{2}
    ]
  9. Limits of exponential and logarithmic functions:
    [
    \lim_{x \to \infty} (1 + \frac{1}{x})^x = e
    ]
    [
    \lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1
    ]

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