Limits

We say that the limit of a function \(f\), as \(x\) approaches \(a\), equals \(L\), and we write

\[ \lim _{x \rightarrow a} f(x)=L \]

provided that the values of \(f(x)\) can be made arbitrarily close to \(L\) by taking \(x\) to be sufficiently close to \(a\). The left-hand and right-hand limits of \(f\), as \(x\) approaches \(a\), are defined similarly:

\[ \lim _{x \rightarrow a^{-}} f(x)=L \quad \lim _{x \rightarrow a^{+}} f(x)=L \]

The limit of \(f\), as \(x\) approaches \(a\), exists if and only if both leftand right-hand limits exist: \(\lim _{x \rightarrow a} f(x)=L\) if and only if \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\).

Algebraic Properties of Limits

The following Limit Laws hold.

  1. \(\lim _{x \rightarrow a}[f(x)+g(x)]=\lim _{x \rightarrow a} f(x)+\lim _{x \rightarrow a} g(x)\)
  2. \(\lim _{x \rightarrow a}[f(x)-g(x)]=\lim _{x \rightarrow a} f(x)-\lim _{x \rightarrow a} g(x)\)
  3. \(\lim _{x \rightarrow a} c f(x)=c \lim _{x \rightarrow a} f(x)\)
  4. \(\lim _{x \rightarrow a}[f(x) g(x)]=\lim _{x \rightarrow a} f(x) \cdot \lim _{x \rightarrow a} g(x)\)
  5. \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}, \quad\) if \(\lim _{x \rightarrow a} g(x) \neq 0\)
  6. \(\lim _{x \rightarrow a}[f(x)]^{n}=\left[\lim _{x \rightarrow a} f(x)\right]^{n}\)
  7. \(\lim _{x \rightarrow a} \sqrt[n]{f(x)}=\sqrt[n]{\lim _{x \rightarrow a} f(x)}\) The following special limits hold.
  8. \(\lim _{x \rightarrow a} c=c\)
  9. \(\lim _{x \rightarrow a} x=a\)
  10. \(\lim _{x \rightarrow a} x^{n}=a^{n}\)
  11. \(\lim _{x \rightarrow a} \sqrt[n]{x}=\sqrt[n]{a}\)

If \(f\) is a polynomial or a rational function and \(a\) is in the domain of \(f\), then \(\lim _{x \rightarrow a} f(x)=f(a)\).

Derivatives

Let \(y=f(x)\) be a function. The derivative of \(\boldsymbol{f}\) at \(\boldsymbol{a}\), denoted by \(f^{\prime}(a)\), is

\[ f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \]

Equivalently, the derivative \(f^{\prime}(a)\) is

\[ f^{\prime}(a)=\lim _{x \rightarrow a} \frac{f(x)-f(a)}{x-a} \]

The derivative of \(f\) at \(a\) is the slope of the tangent line to the curve \(y=f(x)\) at the point \(P(a, f(a))\).

The derivative of \(f\) at \(a\) is the instantaneous rate of change of \(\boldsymbol{y}\) with respect to \(x\) at \(x=a\).

Limits at Infinity

We say that the limit of a function \(f\), as \(x\) approaches infinity, is \(L\), and write

\[ \lim _{x \rightarrow \infty} f(x)=L \]

provided that the values of \(f(x)\) can be made arbitrarily close to \(L\) by taking \(x\) sufficiently large.

We say that the limit of a function \(f\), as \(x\) approaches negative infinity, is \(L\), and we write

\[ \lim _{x \rightarrow-\infty} f(x)=L \]

provided that the values of \(f(x)\) can be made arbitrarily close to \(L\) by taking \(x\) sufficiently large negative. The line \(y=L\) is a horizontal asymptote of the curve \(y=f(x)\) if either

\[ \lim _{x \rightarrow \infty} f(x)=L \quad \text { or } \quad \lim _{x \rightarrow-\infty} f(x)=L \]

The following special limits hold, where \(k>0\) :

\[ \lim _{x \rightarrow \infty} \frac{1}{x^{k}}=0 \quad \text { and } \quad \lim _{x \rightarrow-\infty} \frac{1}{x^{k}}=0 \]

Limits of Sequences

We say that a sequence \(a_{1}, a_{2}, a_{3}, \ldots\) has the limit \(L\), and we write

\[ \lim _{n \rightarrow \infty} a_{n}=L \]

provided that the \(n\)th term \(a_{n}\) of the sequence can be made arbitrarily close to \(L\) by taking \(n\) sufficiently large.

If \(\lim _{x \rightarrow \infty} f(x)=L\) and if \(f(n)=a_{n}\) when \(n\) is an integer, then \(\lim _{n \rightarrow \infty} a_{n}=L\).

Area

Let \(f\) be a continuous function defined on the interval \([a, b]\). The area \(A\) of the region that lies under the graph of \(f\) is the limit of the sum of the areas of approximating rectangles:

\[ \begin{aligned} A & =\lim _{n \rightarrow \infty}\left[f\left(x_{1}\right) \Delta x+f\left(x_{2}\right) \Delta x+\cdots+f\left(x_{n}\right) \Delta x\right] \\ & =\lim _{n \rightarrow \infty} \sum_{k=1}^{n} f\left(x_{k}\right) \Delta x \end{aligned} \]

where

\[ \Delta x=\frac{b-a}{n} \quad \text { and } \quad x_{k}=a+k \Delta x \]

Summation Formulas

The following summation formulas are useful for calculating areas.

\[ \begin{aligned} \sum_{k=1}^{n} c & =n c & \sum_{k=1}^{n} k & =\frac{n(n+1)}{2} \\ \sum_{k=1}^{n} k^{2} & =\frac{n(n+1)(2 n+1)}{6} & \sum_{k=1}^{n} k^{3} & =\frac{n^{2}(n+1)^{2}}{4} \end{aligned} \]