Limit Formula Sheet

A limit describes the value that a function approaches as the input approaches a specified point, and it is fundamental in defining both derivatives and integrals.

Neetesh Kumar | May 31, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Definition
• 2. Left & Right Hand Limit of a Function
• 3. Fundamental Theorems on Limits
• 4. Indeterminate Forms
• 5. General Methods to be Used to Evaluate Limits
• 6. Limit of Trigonometric Functions
• 7. Limit of Exponential Functions
• 8. Limit Using Series Expansion

1. Definition:

Let f(x) be defined on an open interval about ‘a’ except possibly at ‘a’ itself. If f(x) gets arbitrarily close to L (a finite number) for all x sufficiently close to ‘a’, we say that f(x) approaches the limit L as x approaches ‘a’ and we write $\lim\limits_{x \to a} f(x) = L$ and say “the limit of f(x), as x approaches a, equals L”.

2. Left & Right Hand Limit of a Function:

Left hand limit (LHL) = $\lim\limits_{x \to a^-} f(x) = \lim\limits_{h \to 0^+} f(a - h), h>0$
Right hand limit (RHL) = $\lim\limits_{x \to a^+} f(x) = \lim\limits_{h \to 0^+} f(a + h), h>0$
Limit of a function f(x) is said to exist as $x \to a$ when $\lim\limits_{x \to a^-} f(x) = \lim\limits_{x \to a^+} f(x) = L$ (finite and fixed quantity).

Important note:
In $\lim\limits_{x \to a} f(x)$, $x \to a$ necessarily implies $x \ne a$.
That is, while evaluating the limit at $x = a$, we are not concerned with the value of the function at $x = a$. In fact, the function may or may not be defined at $x = a$.
Also, it is necessary to note that if f(x) is defined only on one side of ‘x = a’, one-sided limits are good enough to establish the existence of limits, & if f(x) is defined on either side of ‘a’ both sided limits are to be considered.

3. Fundamental Theorems on Limits:

Let $\lim\limits_{x \to a} f(x) = l$ and $\lim\limits_{x \to a} g(x) = m$. If $l$ and $m$ exist finitely, then:

• Sum rule: $\lim\limits_{x \to a} [f(x) + g(x)] = l + m$
• Difference rule: $\lim\limits_{x \to a} [f(x) - g(x)] = l - m$
• Product rule: $\lim\limits_{x \to a} [f(x)g(x)] = lm$
• Quotient rule: $\lim\limits_{x \to a} \frac{f(x)}{g(x)} = \frac{l}{m}$, provided $m \ne 0$
• Constant multiple rule: $\lim\limits_{x \to a} [kf(x)] = k \cdot \lim\limits_{x \to a} f(x) = kl$
• Power rule: $\lim\limits_{x \to a} [f(x)]^{g(x)} = l^m$, provided $l > 0$
• $\lim\limits_{x \to a} f(g(x)) = f(\lim\limits_{x \to a} g(x)) = f(m)$, provided $f(x)$ is continuous at $x = m$.
• For Example: $\lim\limits_{x \to a} lnf(x) = ln[\lim\limits_{x \to a} f(x)];$ provided lnx is defined at x = $\lim\limits_{t \to a} f(t)$

4. Indeterminate Forms:

$\frac{0}{0}, \frac{\infty}{\infty}, \infty - \infty, 0 \cdot \infty, 1^\infty, 0^0, \infty^0$

Note: Infinity ($\infty$) is a symbol, not a number, and does not obey the laws of elementary algebra.

5. General Methods to be Used to Evaluate Limits:

(a) Factorization: Important factors:

• $x^n - a^n = (x - a)(x^{n-1} + ax^{n-2} + \ldots + a^{n-1})$, $n \in \mathbb{N}$
• $x^n + a^n = (x + a)(x^{n-1} - ax^{n-2} + \ldots + a^{n-1})$, $n$ is an odd natural number.
• $\lim\limits_{x \to a} \frac{x^n - a^n}{x - a} = na^{n-1}$

(b) Rationalization or double rationalization: Rationalize the factor containing the square root and simplify.

(c) Limit when $x \to \infty$:

• Divide by the greatest power of $x$ in the numerator and denominator.
• Put $x = \frac{1}{y}$ and apply $y \to 0$

(d) Squeeze play theorem (Sandwich theorem): If $f(x) \le g(x) \le h(x); \forall$ x and $\lim\limits_{x \to a} f(x) = \lim\limits_{x \to a} h(x) = L$, then $\lim\limits_{x \to a} g(x) = L$.

(e) Using substitution $\lim\limits_{x \to a} f(x) = \lim\limits_{h \to a} f(a-h)$ i.e. by substituting x by a - h or a + h.

6. Limit of Trigonometric Functions:

$\lim\limits_{x \to 0} \frac{\sin x}{x} = 1$, $\lim\limits_{x \to 0} \frac{\tan x}{x} = 1$, $\lim\limits_{x \to 0} \frac{\sin x}{\tan x} = 1$, $\lim\limits_{x \to 0} \frac{\tan x}{\sin x} = 1$.

If $\lim\limits_{x \to a} f(x) = 0$, then $\lim\limits_{x \to a} \frac{\sin f(x)}{f(x)} = 1$.

7. Limit of Exponential Functions:

(a) $\lim\limits_{x \to 0} (1 + x)^{\frac{1}{x}} = e = \lim\limits_{x \to \infty} (1 + \frac{1}{x})^x$
In general, if $\lim\limits_{x \to a} f(x) = 0$, then $\lim\limits_{x \to a} (1 + f(x))^{\frac{1}{f(x)}} = e$

(b) If $\lim\limits_{x \to a} f(x) = A > 0$ and $\lim\limits_{x \to a} \phi(x) = B$ (finite quantity), then $\lim\limits_{x \to a} [f(x)]^{\phi(x)} = e^{BlnA} = A^B$

(c) $\lim\limits_{x \to 0} \frac{a^x - 1}{x} = lna$ (a > 0) In particular $\lim\limits_{x \to 0} \frac{e^x - 1}{x} = 1$
In general If $\lim\limits_{x \to 0} f(x) = 0,$ then $\lim\limits_{x \to a} \frac{a^{f(x)} - 1}{f(x)} = lna,$ a > 0

(d) $\lim\limits_{x \to 0} \frac{ln(1+x)}{x} = 1$

(e) $\lim\limits_{x \to a} f(x) = 1$ and $\lim\limits_{x \to a} \phi(x) = \infty$ then $\lim\limits_{x \to a} [f(x)]^{\phi(x)} = e^k$ where k = $\lim\limits_{x \to a} \phi(x)[f(x)-1]$

8. Limit Using Series Expansion:

Remember these series expansions:
(a) $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots, x \in R$

(b) $a^x = 1 + \frac{xlna}{1!} + \frac{x^2ln^2a}{2!} + \frac{x^3ln^3a}{3!} + ..., a>0$

(c) $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \ldots$

(d) $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \ldots$

(e) $tanx = x + \frac{x^3}{3} + \frac{2x^5}{15} + \ldots, |x| < \frac{\pi}{2}$

(f) $tan^{-1}x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} \ldots, x\in R$

(g) $sin^{-1}x = x + \frac{1^2x^3}{3!} + \frac{1^2.3^2x^5}{5!} + \frac{1^2.3^2.5^2x^7}{7!} \ldots, |x| \le 1$

(h) $\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \ldots$ for $-1 < x \le$ 1

(i) $(1 + x)^n = 1 + nx + \frac{n(n-1)x^2}{2!} + \ldots, n \in R, |x| < 1$

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