Functions: \(f\left( x \right),\) \(g\left( x \right)\)

Argument (independent variable): \(x\)

Argument (independent variable): \(x\)

Real constant numbers: \(a,\) \(k,\) \(L,\) \(\varepsilon,\) \(\delta\)

- A function \(f\left( x \right)\) has the limit \(L\) as \(x\) approaches \(a\) if, for every positive number \(\varepsilon\), there exists a positive number \(\delta\) such that \(\left| {f\left( x \right) – L} \right| \lt \varepsilon \) whenever \(0 \lt \left| {x – a} \right| \lt \delta \). The limit of a function is denoted by

\(\lim\limits_{x \to \infty } f\left( x \right) = L\). - The limit of a constant is equal to the constant:

\(\lim\limits_{x \to a} C = C\) - The limit of the sum of two functions is equal to the sum of their limits (assuming these limits exist – this remark applies also to other formulas below):

\(\lim\limits_{x \to a} \left[ {f\left( x \right) + g\left( x \right)} \right] =\) \( \lim\limits_{x \to a} f\left( x \right) \,+\) \( \lim\limits_{x \to a} g\left( x \right)\) - The limit of the difference of two functions is the difference of their limits:

\(\lim\limits_{x \to a} \left[ {f\left( x \right) – g\left( x \right)} \right] =\) \(\lim\limits_{x \to a} f\left( x \right) \,-\) \(\lim\limits_{x \to a} g\left( x \right)\) - The limit of the product of two functions is equal to the product of their limits:

\(\lim\limits_{x \to a} \left[ {f\left( x \right) \cdot g\left( x \right)} \right] =\) \( \lim\limits_{x \to a} f\left( x \right) \cdot \lim\limits_{x \to a} g\left( x \right)\) - The limit of the quotient of two functions is the quotient of their limits, provided the limit in the denominator is non-zero:

\(\lim\limits_{x \to a} {\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} =\) \({\large\frac{{\lim\limits_{x \to a} f\left( x \right)}}{{\lim\limits_{x \to a} g\left( x \right)}}\normalsize},\) if \(\lim\limits_{x \to a} g\left( x \right) \ne 0.\) - A constant factor can be taken out of the limit:

\(\lim\limits_{x \to a} \left[ {kf\left( x \right)} \right] =\) \( k\lim\limits_{x \to a} f\left( x \right)\) - Limit of a composite function

\(\lim\limits_{x \to a} f\left( {g\left( x \right)} \right) =\) \( f\left( {\lim\limits_{x \to a} g\left( x \right)} \right)\) - Limit of a continuous function

If the function \(f{\left( x \right)}\) is continuous at \(x=a,\) then

\(\lim\limits_{x \to a} f\left( x \right) = f\left( a \right)\). - \(\lim\limits_{x \to 0} {\large\frac{{\sin x}}{x}\normalsize} = 1\)
- \(\lim\limits_{x \to 0} {\large\frac{{\tan x}}{x}\normalsize} = 1\)
- \(\lim\limits_{x \to 0} {\large\frac{{\arcsin x}}{x}\normalsize} = 1\)
- \(\lim\limits_{x \to 0} {\large\frac{{\arctan x}}{x}\normalsize} = 1\)
- \(\lim\limits_{x \to 0} {\large\frac{{\ln\left( {1 + x} \right)}}{x}\normalsize} = 1\)
- \(\lim\limits_{x \to \infty } {\left( {1 + {\large\frac{1}{x}}\normalsize} \right)^x} = e\)
- \(\lim\limits_{x \to \infty } {\left( {1 + {\large\frac{k}{x}}\normalsize} \right)^x} = e^k\)
- \(\lim\limits_{x \to 0} {a^x} = 1\)