Table of Content

• What is a Limit in Mathematics?
• Limit Formulas
• Basic Limit Formulas
• Trigonometric Limits
• L-hospital Rule
• Exponential Limits
• Logarithmic limits
• Important Limit Results
• Sample Problem
• Practice Problems
• FAQs

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

Some Shortcut Formulas

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

Solution:

lim_{x ⇢ ∞} (1 + [Tex]\frac{sinx}{x}        [/Tex])

1 + lim_{x ⇢ ∞}[Tex] \frac{sinx}{x} [/Tex]

As we know, x = ∞  

So 1/x = 0

[Tex]1 + lim\frac{1}{x}⇢∞ \frac{sin\frac{1}{x}}{x} [/Tex]

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}  [Tex]\frac {e^x -(1+x+ \frac {x^2}{2})}{ x^3} [/Tex]

Solution:

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

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

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

Solution:

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

lim_{a ⇢ 0}  x^alogx = logx

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

Solution:

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

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)]

FAQs on Limit Formula

What is a limit in calculus?

A limit in calculus describes the value that a function approaches as the input (or variable) approaches a certain point.

How do you evaluate a limit?

Limits can be evaluated using various methods, such as direct substitution, factoring, rationalizing, using special limit formulas, and applying the squeeze theorem or L’Hôpital’s rule.

What are some basic limit formulas?

Some basic limit formulas include: