Limits

If $f(x)$ is defined for all $x$ near a, except possibly at a itself, and if it can be ensured that $f(x)$ is as close to $L$ by taking $x$ close enough to $a$, but not equal to $a$. Then $f$ approaches the limit $L$ as $x$ approaches $a$:

\[ \lim_{x \to a} f(x) = L \]

One-sided limits

If $f(x)$ is defined on some interval $(b,a)$ extending to the left of $x=a$, and if it can be ensured that $f(x)$ is as close to $L$ by taking $x$ to the left of $a$ and close enough to $a$, then $f(x) has left limit $L$ at $x=a$ and:

\[ \lim_{x \uparrow a} f(x) = L. \]

If $f(x)$ is defined on some interval $(b,a)$ extending to the right of $x=a$ and if it can be ensured that $f(x)$ is as close to $L$ by taking $x$ to the right of $a$ and close enough to $a$, then $f(x) has right limit $L$ at $x=a$ and:

\[ \lim_{x \downarrow a} f(x) = L. \]

Limits at infinity

If $f(x)$ is defined on an interval $(a,\infty)$ and if it can be ensured that $f(x)$ is as close to $L$ by taking $x$ large enough, then $f(x)$ approaches the limit $L$ as $x$ approaches infinity and

\[ \lim_{x \to \infty} f(x) = L \]

Limit rules

If $\lim_{x \to a} f(x) = L$, $\lim_{x \to a} g(x) = M$, and $k$ is a constant then,

Limit of a sum: $\lim_{x \to a}[f(x) + g(x)] = L + M$.
Limit of a difference: $\lim_{x \to a}[f(x) - g(x)] = L - M$.
Limit of a multiple: $\lim_{x \to a}k f(x) = k L$.
Limit of a product: $\lim_{x \to a}f(x) g(x) = L M$.
Limit of a quotient: $\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{L}{M}$, if $M \neq 0$.
Limit of a power: $\lim_{x \to a}[f(x)]^\frac{m}{n} = L^{\frac{m}{n}}$.

Formal definition of a limit

The limit $\lim_{x \to a} f(x) = L$ means,

\[ \forall \varepsilon_{> 0} \exists \delta_{>0} \Big[ 0<|x-a|<\delta \implies |f(x) - L| < \varepsilon \Big]. \]

The limit $\lim_{x \to \infty} f(x) = L$ means,

\[ \forall \varepsilon_{> 0} \exists N_{>0} \Big[x > N \implies |f(x) - L | < \varepsilon \Big]. \]

The limit $\lim_{x \to a} f(x) = \infty$ means,

\[ \forall M_{> 0} \exists \delta_{>0} \Big[ 0<|x-a|<\delta \implies f(x) > M \Big]. \]

The limit $\lim_{x \to \infty} f(x) = \infty$ means,

\[ \forall M_{> 0} \exists N_{>0} \Big[ x > N \implies f(x) > M \Big]. \]

For one-sided limits there are similar formal definitions.

Example

Applying the formal definition of a limit for $\lim_{x \to 4}\sqrt{2x + 1}$

Given $\varepsilon > 0$
Choose $\delta = \frac{\varepsilon}{2}$
Suppose $0 < |x - 4| < \delta$
Check $|\sqrt{2x + 1} - 3|$

\[ \begin{array}{ll} |\sqrt{2x + 1} - 3| &= |\frac{(\sqrt{2x + 1} - 3)(\sqrt{2x + 1} + 3)}{\sqrt{2x + 1} + 3}|\\ &= \frac{2|x - 4|}{\sqrt{2x + 1} + 3}\\ &< 2|x-4|\\ &< 2\delta = \varepsilon \end{array} \]

Squeeze Theorem

Suppose that $f(x) \leq g(x) \leq h(x)$ holds for all $x$ in some open interval containing $a$, except possibly at $x=a$ itself. Suppose also that

\[\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L.\]

Then $\lim_{x \to a} g(x) = L$ also. Similar statements hold for left and right limits.

Example

Applying squeeze theorem on $\lim_{x \to 0} x^2 \cos(\frac{1}{x})$.

\[ \begin{array}{ll} \forall x \neq 0\\ -1 \leq \cos(\frac{1}{x}) \leq 1 \implies -x^2 \leq x^2 \cos(\frac{1}{x}) \leq x^2\\ \mathrm{Since,} \space \lim_{x \to 0} x^2 = \lim_{x \to 0} -x^2 = 0\\ \lim_{x \to 0} x^2 \cos(\frac{1}{x}) = 0 \end{array} \]