Limit Formula

Limits help us comprehend how functions behave as their inputs approach certain values. Think of a limit as the destination that a function aims to reach as the input gets closer and closer to a specific point.

In this article, we will explore the essential limit formulas that form the backbone of calculus. These formulas are like the rules of a game, guiding us on how to find limits in various scenarios. Whether you're adding, subtracting, multiplying, or dividing functions, there are specific formulas to help you determine the limit.

What is a Limit in Mathematics?

A limit is the value that a function (or sequence) "approaches" as the input (or index) approaches some value.

Mathematically, for a function f(x), the limit of f(x) as approaches c is L, written as:

lim⁡_x→cf(x) = L

This means that for every ϵ > 0, there exists a δ>0 such that whenever 0 < ∣x − c∣ < δ, it follows that ∣f(x) − L∣ < ϵ.

Read More about Formal Definition of Limits.

Limit Formulas

There are various formulas involving the limit such as:

• Basic Limit Formulas
• Trigonometric Limits
• L-hospital Rule
• Exponential Limits
• Logarithmic Limits

Let's discuss these formulas in detail as follows:

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.  

• lim_{x ⇢ 0}  \frac{sinx}{x}  = lim_{x ⇢ 0} \frac{x}{sinx}  = 1
• lim_{x ⇢ 0} tanx/x = lim_{x ⇢ 0} x/tanx =1

As we considered our first one, 

lim_{x ⇢ 0} sinx/x =1      

Using L-Hospital

lim_{x ⇢ 0} cosx/1 

lim_{x ⇢ 0} cos(0)/1 = 1/1 =1

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

Indeterminate Form

0/0, ∞/∞, ∞-∞, ∞/0, 0^∞, ∞⁰ , 0⁰, ∞^∞

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} sin^-1x/x =1

lim_{x ⇢ 0} 1/√1+x²  [Using L-Hospital] 

= 1/√(1 + (0)²) = 1

• lim_x⇢0 \frac{tan^{-1}x}{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 ⇢ a} sin(x - a) / (x - a) 

=1

lim_{x ⇢ a} cos(x - a)/1 

= lim_{x ⇢ a} cos(a - a) = cos(0) =1

• lim_x⇢∞ sinx/x = 0
• lim_x⇢∞ cosx/x = 0
• lim_x⇢∞ sin(1/x) / (1/x) =0

lim_{x ⇢ ∞} sin(1/x)/(1/x) = 0

Let 1/x = h

So, limits changes to 0

Because 1\∞ = 0

lim_{h ⇢ 0} sinh/h

As we see before, If lim_{x ⇢ 0} sinx/x = 1

So, lim_{h ⇢ 0} sinh/h = 1

Read More about L'Hospital Rule.

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

Read More about Exponents.

Logarithmic Limits

Some of the common formula related to logarithmic 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

Read More about Logarithm.

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}
3. lim_x⇢a f(x)^g(x) = e^{limx\to a\left[ f(x)-1 \right].g(x)}

Conclusion

Limit Formula is a powerful concept that bridges the gap between algebra and calculus, providing a stepping stone to advanced mathematical analysis. By mastering the limit formula, students can better understand the continuity, behavior, and trends of functions. These formulas are crucial for solving complex problems in calculus and for further studies in mathematics.

Read More,

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

Sample Problem on Limit Formula

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

Solution:

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.

Solution:

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

Solution:

Using L-hospital 

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

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

Solution:

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 = 0                                                                                                                                                 

Question 5: Solve, lim_{x ⇢ π/2} (tanx)^cosx 

Solution:

let Y = lim_{x ⇢ π/2}  (tanx)^cosx

Taking log_e both sides,

 log_eY = lim_{x ⇢ π/2}  log_e(tanx)^cosx 

 log_eY = lim_{x ⇢ π/2}  cosx log_e(tanx)

log_ey = lim_{x ⇢ π/2} loge(tanx)/secx

Using l-hospital,

log_ey = lim_{x ⇢ π/2} cosx /sin²x = 0

Now, taking exponent on both sides,

Y = lim_{x ⇢ π/2} e⁰ 

Y = lim_{x ⇢ π/2}  (tanx)^cosx = 1  

Question 6: lim_{x ⇢ 0}  \frac {e^x -(1+x+ \frac {x^2}{2})}{ x^3}

Solution:

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

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

Question 7: Solve, lim_{a ⇢ 0}  \frac{x^a-1}{a}

Solution:

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

lim_{a ⇢ 0}  x^alogx = logx

Question 8: Solve, lim_{x ⇢ 0} \frac{x^2+x-sinx}{x^2}

Solution:

lim_{x ⇢ 0} \frac{x^2+x-(x-x^3/3!)}{x^2}

lim_{x ⇢ 0} 1 + x/3! = 1

Practice Problems on Limit Formula

Problem 1: 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)]