Table of Contents

Formulas, Rules and Theorems of Limits of Functions

Definition

1) Formal Definition ("epsilon-delta definition")
The limit of \( f(x) \), as \( x \) approaches \( a \) , exists and is equal to \( L \) written as \[ \displaystyle \lim_{x\to a} f(x) = L \] if for any value of \( \epsilon \gt 0 \) we can find a value of \( \delta \gt 0 \)such that if \( 0 \lt |x - a| \lt \delta \) then \( |f(x) - L| \lt \epsilon \)
2) Working Definition
The limit of \( f(x) \), as \( x \) approaches \( a \) , exists and is equal to \( L \) written as \[ \displaystyle \lim_{x\to a} f(x) = L \] if we can make the values of \( f(x) \) as close as we want to \( L \) as \( x \) takes values closer to and on either sides of \( a \).
Note that the function may or may not be defined at \( x = a \) for a limit of a function to exist at \( x = a \).

Formulas of Limits

• \( \displaystyle \lim_{x\to 0} \dfrac{\sin x}{x} = 1 \) (See Squeezing Theorem to Find Limits of Mathematical Functions for proof)
• \( \displaystyle \lim_{x\to 0} \dfrac{\sin^{-1} x}{x} = 1 \)
• \( \displaystyle \lim_{x\to 0} \dfrac{\tan x}{x} = 1 \)
• \( \displaystyle \lim_{x\to 0} \dfrac{\tan^{-1} x}{x} = 1 \)
• \( \displaystyle \lim_{x\to 0} \dfrac{1 - \cos x}{x} = 0 \) (See Limits of Trigonometric Functions for proof)
• \( \displaystyle \lim_{x\to 0} \dfrac{a^x - 1}{x} = \ln a \) , if \( a \gt 0 \)
• \( \displaystyle \lim_{x\to 0} \dfrac{e^x - 1}{x} = 1 \)
• \( \displaystyle \lim_{x\to \infty} (1+\dfrac{1}{x})^x = e \) (See Euler Constant e for definition of e)
• \( \displaystyle \lim_{x\to \infty} (1+\dfrac{1}{x})^{kx} = e^k \)
• \( \displaystyle \lim_{x\to 0} (1+x)^{1/x} = e \)
• \( \displaystyle \lim_{x\to 0} \dfrac{\ln(1+x)}{x} = 1 \)
• \( \displaystyle \lim_{x\to 0^+} \log_b x = -\infty \) , for \( b \gt 1 \)
• \( \displaystyle \lim_{x\to \infty} \log_b x = \infty \) , for \( b \gt 1 \)
• \( \displaystyle \lim_{x\to 0^+} \log_b x = \infty \) , for \(0 \lt b \lt 1 \)
• \( \displaystyle \lim_{x\to \infty} \log_b x = -\infty \) , for \( 0 \lt b \lt 1 \)
• \( \displaystyle \lim_{x\to +\infty} b^x = \infty \) , for \( b \gt 1 \)
• \( \displaystyle \lim_{x\to - \infty} b^x = 0 \) , for \( b \gt 1 \)
• \( \displaystyle \lim_{x\to +\infty} b^x = 0 \) , for \( 0 \lt b \lt 1 \)
• \( \displaystyle \lim_{x\to - \infty} b^x = \infty \) , for \( 0 \lt b \lt 1 \)
• \( \displaystyle \lim_{x\to + \infty} x^{\frac{1}{x}} = 1 \)
  Note that many of the limits of ratios listed above, can be found using the L'Hôpital's Rule.

Theorems and Rules of Limits

Let \( \displaystyle \lim_{x\to a} f(x) = L_1 \) and \( \displaystyle \lim_{x\to a} g(x) = L_2 \)
• \( \displaystyle \lim_{x\to a} (f(x) \pm g(x)) = \displaystyle \lim_{x\to a} f(x) \pm \displaystyle \lim_{x\to a} g(x) = L_1 \pm L_2\)
• \( \displaystyle \lim_{x\to a} (f(x) \cdot g(x)) = \displaystyle \lim_{x\to a} f(x) \cdot \displaystyle \lim_{x\to a} g(x) = L_1 \cdot L_2\)
• \( \displaystyle \lim_{x\to a} (k \cdot g(x)) = k \cdot \displaystyle \lim_{x\to a} f(x) = k \cdot L_1 \) , where \( k \) is a constant
• \( \displaystyle \lim_{x\to a} (\dfrac{f(x)}{g(x)}) = \dfrac{ \displaystyle \lim_{x\to a} f(x)}{ \displaystyle \lim_{x\to a} g(x) } = \dfrac{L_1}{L_2} \) if \( L_2 \ne 0 \)
• \( \displaystyle \lim_{x\to a} [f(x)]^n = [ \displaystyle \lim_{x\to a} f(x) ]^n = L_1^n \) , if \( n \) is a positive integer.
• \( \displaystyle \lim_{x\to a} \sqrt[n]{f(x)} = \sqrt[n] { \displaystyle \lim_{x\to a} {f(x)}} = \sqrt[n]{L_1} \) ; if \( n \) is a positive integer and \( L_1 \gt 0 \) If \( n \) even ,
• If \( f(x) \) is a continuous function, then \( \displaystyle \lim_{x\to a} f(x) = f(a) \)
  
  
• Squeeze Theorem (or Sandwich Theorem)
  If \( f(x) \le g(x) \le h(x) \) for \( x \) near \( a \) and if \( \displaystyle \lim_{x\to a} f(x) = \displaystyle \lim_{x\to a} h(x) = L \) , then \( \displaystyle \lim_{x\to a} g(x) = a \)
  
  
• L'Hôpital's Rule
  If \( \displaystyle \lim_{x\to a} f(x) = 0 \) and \( \displaystyle \lim_{x\to a} g(x) = 0 \)
  or \( \displaystyle \lim_{x\to a} f(x) = +\infty \) and \( \displaystyle \lim_{x\to a} g(x) = +\infty \)
  or \( \displaystyle \lim_{x\to a} f(x) = -\infty \) and \( \displaystyle \lim_{x\to a} g(x) = -\infty \)
  and if \( \displaystyle \lim_{x\to a} \dfrac{f(x)}{g(x)} \) exists , then
  \( \quad \quad \quad \displaystyle \lim_{x\to a} \dfrac{f(x)}{g(x)} = \displaystyle \lim_{x\to a} \dfrac{f'(x)}{g'(x)} \)

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