Formulas, Rules, and Theorems for 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 every 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) Functional 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 both sides of \( a \).
Note that the function may or may not be defined at \( x = a \) for a function limit to exist at \( x = a \).
Limit Formulas
- \( \displaystyle \lim_{x\to 0} \dfrac{\sin x}{x} = 1 \) (See Squeeze Theorem for Finding Limits of Mathematical Functions for the 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 the 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's Constant e for the 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 ratio limits listed above can be found using L'Hôpital's Rule.
Limit Theorems and Rules
-
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 \( n \) is 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 \( \displaystyle \lim_{x\to a} g(x) = L \)
- 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
\( \quad \quad \quad \displaystyle \lim_{x\to a} \dfrac{f(x)}{g(x)} = \displaystyle \lim_{x\to a} \dfrac{f'(x)}{g'(x)} \)
More References and Links
Introduction to Limits in Calculus
L'Hôpital's Rule
Squeeze Theorem
Continuous Functions in Calculus
List of limits
Joel Hass, University of California, Davis; Maurice D. Weir Naval Postgraduate School; George B. Thomas, Jr. Massachusetts Institute of Technology; University Calculus, Early Transcendentals, Third Edition, Boston Columbus, 2016, Pearson.
Gilbert Strang; MIT, Calculus, Wellesley-Cambridge Press, 1991
Mathematics for Engineers with Examples and Solutions