Limit Formulas

Formulas make it easier and faster to evaluate limits without having to use lengthy calculations every time. For example, instead of deriving the result every time, we can directly use the standard formula:

\lim_{x \to 0} \frac{x}{\sin x} = 1

There are various formulas involving the limit, such as:

Basic Limit Formulas

Trigonometric Limits

To evaluate trigonometric limits, we have to reduce the terms of the function into simpler terms or into terms of sinθ and cosθ.  

\begin{aligned}&\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad\lim_{x \to 0} \frac{x}{\sin x} = 1 \\[6pt]&\lim_{x \to 0} \frac{\tan x}{x} = 1, \quad\lim_{x \to 0} \frac{x}{\tan x} = 1 \\[6pt]&\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}, \quad\lim_{x \to 0} \frac{\cos x - 1}{x^2} = -\frac{1}{2} \\[6pt]&\lim_{x \to a} \frac{\sin x - \sin a}{x - a} = \cos a, \quad\lim_{x \to a} \frac{\cos x - \cos a}{x - a} = -\sin a \\[6pt]\end{aligned}

If the function gives an indeterminate form by putting limits, Then use the L'hospital rule.

L'hospital Rule

If we get the indeterminate form, then we differentiate the numerator and denominator separately until we get a finite value. Remember we would differentiate the numerator and denominator the same number of times. Similarly for all trigonometric function,

• lim_{x ⇢ 0} sin^-1x/x = lim_{x ⇢ 0} x/sin^-1x = 1
• lim_x⇢0 tan^-1x/x=1
• lim_{x ⇢ a}  sin x^o/x = π/180
• lim_x⇢0 cosx = 1
• lim_{x ⇢ a}  sin(x-a) / (x-a) =1
• lim_x⇢∞ sinx/x = 0
• lim_x⇢∞ cosx/x = 0
• lim_x⇢∞ sin(1/x) / (1/x) =0

Exponential Limits

Some of the common formula related to exponents are:

• lim_{x ⇢ 0}  e^x - 1 /x = 1
• lim_{x ⇢ 0}  a^x - 1 /x = log_ea
• lim_{x ⇢ 0} e^λx - 1 /x = λ

Logarithmic Limits

Some of the common formula related to logarithm limits are:

• lim_{x ⇢ 0} log(1 + x) /x = 1
• lim_{x ⇢ e} log_ex = 1
• lim_{x ⇢ 0} log_e(1 - x) /x  = -1
• lim_{x ⇢ 0} log_a(1 + x) /x = log_ae

Important Limit Results

Some of the important results involving limits are:

1. lim_x⇢0 (1+x)^{\frac{1}{x}}  = e
2. lim_x⇢0 \left( 1+\frac{1}{x} \right)^x  = e
3. lim_x⇢0 \frac{e^x-1}{x} =1
4. lim_x⇢0\frac{a^x-1}{x}  = log_ea
5. lim_x⇢0 \frac{1-cosmx}{x^2}  = m²/2
6. lim_x⇢0 \frac{1-cosmx}{1-cosnx} = \frac{m^2}{n^2}

Some Shortcut Formulas

1. lim_x⇢0 \left( 1+ \frac{a}{b}x \right)^\frac{c}{dx} = e^\frac{ac}{bd}
2. lim_x⇢0 \left( 1+ \frac{a}{bx} \right)^\frac{cx}{d} = e^\frac{ac}{bd}

Related Articles

• Limits
• Differentital Calculus
• Calculus Cheat Sheet
• Symbols in Calculus

Sample Problem

Question 1: Solve, lim_x⇢0  (x - sinx ) /(1 - cosx).

Using L-hospital,

lim_{x ⇢ 0} (1 - cosx) / (sinx)

lim_{x ⇢ 0} sinx / cosx = sin(0) / cos(0) = 0/1 = 0

Question 2: Solve, lim_{x ⇢ 0} (e^2x -1) / sin4x.

Using L-hospital 

lim_{x ⇢ 0} (2)(e^2x) / cos4x

lim_{x ⇢ 0}  2(e⁰) / cos4(0) = 2/1= 2

Question 3: Solve, lim_{x ⇢ 0} (1 - cosx) / x²

Using L-hospital 

lim_{x ⇢ 0} sinx /2x = 1/2 {sinx/x = 1}

Question 4: Solve, lim_{x ⇢ ∞} \frac{x+sinx}{x}

lim_{x ⇢ ∞} (1 + \frac{sinx}{x}        )

1 + lim_{x ⇢ ∞} \frac{sinx}{x}

As we know, x = ∞  

So 1/x = 0

1 + lim\frac{1}{x}⇢∞ \frac{sin\frac{1}{x}}{x}

1 + 0 = 1                                                                                                                                                

Question 5: lim_{x ⇢ 0}  \frac {e^x -(1+x+ \frac {x^2}{2})}{ x^3}

lim_x⇢0 \frac{1+\frac{x}{1!} + \frac{x2}{2!} + \frac{x3}{3!} - ( 1+ x+ \frac{x2}{2!} ) }{x3}

lim_x⇢0 \frac{\frac{x3}{3!}}{x3}  = 1/3! =1/6

Question 6: Solve, lim_{a ⇢ 0}  \frac{x^a-1}{a}

Using L-hospital (Differentiating numerator and denominator w.r.t a)  

lim_{a ⇢ 0}  x^alogx = logx

Practice Problems

Simplify:

• lim_x→2(3x + 4)
• lim_x→-1(x² + 2x + 1)
• lim_x→0(sin²x/2x)
• lim_x→3[(x² - 9)/(x - 3)]
• lim_x→∞[(5x + 7)/(2x - 3)]
• lim_x→0[(1 - cos x)/x]
• lim_x→-∞(2x³ − x² + x)
• lim_x→∞(e^-x)
• lim_x→0[(e^x - 1)/x]
• lim_x→∞[(2x² − 5x + 3​)/(x - 3)]