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L’Hopital’s Rule & Indeterminate Forms

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4 months ago

December 26, 2024

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L’Hopital’s Rule deals with indeterminate form situations where direct substitution of values into a limit expression does not provide a clear answer.

What is L’Hopital’s Rule?

L’Hopital’s Rule allows you to evaluate limits of indeterminate forms like 0/0 and ∞/∞. These shapes frequently appear when we attempt to calculate the limit of a ratio of two functions. Instead of immediately attempting to simplify the formula, L’Hopital’s Rule permits us to differentiate the numerator and denominator independently before reevaluating the limit.

If a limit results in an indeterminate form 0/0 or ∞/∞, then:

Where:

f(x) and g(x) represent the functions in the numerator and denominator, respectively.
The derivatives of these functions are f ′ (x) and g′ (x) respectively.

The procedure can be repeated, taking derivatives until a determinate form is obtained, if the limit on the right-hand side continues to give an indeterminate form.

When to Apply L’Hopital’s Rule?

L’Hopital’s formula applies to the following cases:

0/0 Indeterminate Form: If inserting the limit point into the numerator and denominator gives 0, you can use L’Hopital’s Rule.

Example:

(both numerator and denominator approach 0)

∞/∞ Indeterminate Form: L’Hopital’s Rule applies if, as x approaches the limit, both the numerator and denominator go towards infinity.

Example:

(both numerator and denominator approach ∞)

How to Apply L’Hopital’s Rule:
Let’s analyze how to solve limits using L’Hopital’s Rule.

Identify the Indeterminate Form First: make sure that an indeterminate form, such as 0/0 or ∞/∞, is produced by direct substitution into the limit expression.
Differentiate the Numerator and Denominator: Compute the derivatives of the functions in the numerator and denominator separately.
Reevaluate the Limit: Enter the limit value once more after distinguishing. Repeat the procedure if the outcome is still undetermined.
Repeat if necessary:
If the resulting expression is still indeterminate, apply L’Hopital’s Rule again by differentiating the new numerator and denominator.

Example 1: Evaluating a Limit
To demonstrate L’Hopital’s formula in action, let’s look at a straightforward scenario.

For instance:

L'Hopital's Rule

Direct substitution: Substituting $x = 0$ into the expression, we get

an indeterminate form.

2. Differentiate: Distinguish between the denominator and numerator.

Cos(x) is the derivative of sin(x).
1 is the derivative of x.

3. Re-evaluate the limit: Now, we have:

L'Hopital's Rule

Thus the limit is 1.

Example 2: Using L’Hopital’s formula for ∞/∞ Form

Consider the limit:

1. Direct substitution: Substituting

which is an indeterminate form

2.Differentiate:

- The derivative of $x^2$ is
- The derivative of $e^x is e^x.$

3. Reevaluate the limit:
L’Hopital’s Rules is applied, and the result is:

. So we apply L’Hopital’s Rules once more.

4. Again differentiate:

2 is the derivative of 2x.
e^x is the derivative of e^x.

5. Re-evaluate the limit:
The expression now becomes:

As , $e^x$ grows exponentially, and the limit of the expression is 0.

Thus, the limit is 0.

When L’Hopital’s Rule Should Not Be Used
This is effective, but it’s not always the greatest approach. In the following situations, it is not applicable:

This is not required if the limit does not produce an indeterminate form (such as 0/5, 5/0, or other determinate forms).
You might want to look at alternative methods, such as factoring or simplifying the function, if using the rule results in a more complicated expression.

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All formulas of circle-Concepts , Properties

Published

2 months ago

March 1, 2025

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All formulas of circle

All formulas of circle are essential concepts in geometry.

Definition

A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius (r).

Here

Diameter: The diameter is twice the radius: D = 2r.
Circumference: The circumference (C) is the total distance around the circle: $C = 2 π r$
Area: The area enclosed by the circle is

Chord: A chord is a line segment joining two points in the circle.
Arc: An arc is a part of the circle’s circumference.
Sector: A sector is a region enclosed by two radii and an arc.

Standard equation of a circle

The equation of circle depends on its center and radius

(a) Circle centered at the origin (0,0)

If the center is at the origin and the radius is r, then the equation is:

(b) Circle centered at (h,k)

If the center is at (h, k), and the radius is r, the equation is :

General equation of a circle

The expanded form of a circle’s equation is :

The center is given by (-g,-f).
The radius is

Important properties of circle

1. Tangents and normals to a circle

A tangent is a line that touches the circle at exactly one point.The tangent to the circle equation

In a general equation, x^2 +y^2 +2gx+2fy+c=0 , the equation of the tangent at (x_1,y_1) is:

2. Polar form of a circle: In polar coordinates (r,θ), the equation of a circle centered at the origin is

3. Position of any points with respect to a circle:

Distance formula: Calculate the distance d of the point (x_1,y_1) from the center (h,k):

Now, compare d with the radius r:

If d<r: The point is inside the circle.
If d=r : The point is on the circle.
If d>r : The point is outside the circle.

4. Length of tangent from the point (x_1, y_1) to the circle: The length of the tangent from an external point $(x_1, y_1) to$ a circle with equation

Example

Given circle

Find the tangent length from (7,1)

Using the formula

5. Angle of Intersection of two circles

The angle of intersections between two circles refers to the angle between their tangents at the points of intersection.It can be calculated using the formula

All formulas of circle

Example

Given two circles

(x-2)^2+(y-3)^2=25 – Center(2,3), Radius r_1=5

(x-7)^2+(y-6)^2=16 -Center(7,6), Radius r_2=4

Step 1: Find distance between centers

Step 2: Apply the formula

Step 3: Find θ

All formulas of circle

Special cases

1. Orthogonal circles ( θ=90∘)

Two circles are orthogonal (intersect at right angles) if their centers and radii satisfy

This means and

2. Concentric circles

If two circles share the same center (d = 0), they do not intersect or are the same circle .

3. Circles touching externally (θ=0∘)

If d=r1+r2, then they touch externally.

4. Circles touching internally ()

If d=∣r1−r2∣ they touch internally.

These are All formulas of circle.

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How to make math fun through storytelling: Chapter 3

Published

2 months ago

February 25, 2025

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How to make math fun through storytelling: The secret of the shadow pyramid

How to make math fun through storytelling: The math explorers cautiously stepped into a long hall, but something was off.The floor was covered in golden sand, and that moved in waves shifting unpredictably beneath their feet .The walls were lined with ancient measuring tools—rulers, scales, and hourglasses—glowing faintly with magical energy.

Chapter 3: The Shifting Sands of Measurement

Joey took a step forward, but the sand beneath him suddenly crumbled, nearly swallowing his foot.”Whoa! The ground isn’t stable, he gasped. Max pointed to an inscription on the wall.”Look ! A puzzle!Maybe it will help us cross.”

The inscription read: “To find the safe path, convert and compare:

A rope is 2.5 meters long . How many centimeters is that?
One stone weighs 3,000 grams. “How many kilograms is that?”

Ross grinned. “We just need to convert the units!”

“We know that 1 meter equals 100 centimeters.” Max said, writing on the sand with her finger .”So 2.5 meters is…”

2.5×100=250 cemtimeters.

“And for the weight ,”joey continued . “1 kilogram is 1000 grams.So we divide 3000 by 1000.”

3000÷1000=3 kilograms .

As soon as they said the answers aloud, the numbers curved into the walls began to glow . The sand shifted, revealing solid stepping stones that created a safe path forward .

“We did it!” Max cheered. “The numbers unlocked the way!”

One by one, the explorers carefully stepped onto the stones, making their way across the hallway.But as they reached the middle, another inscription appeared on a stone piller:

“To continue forward, solve these conversions:

A river is 5 kilometers long.How many meters is that ?
A bottle holds 2.5 liters of water. How many milliliters is that?
A clock ticks every 15 minutes.How many ticks occur in 2 hours?

“We can do this !” said Ross, thinking quickly.

5 kilometers×1000=5000 meters.
2.5 liters×1000=2500 milliliters.
2 hours = 120 minutes.120÷15=8 ticks

Once they spoke the correct answers, the stepping stones ahead lit up, revealing the rest of the safe path .The shifting sand behind them swallowed the remaining unstable floor, making it impossible to turn back. “Looks like we solved it just in time,” Joey said with a relieved smile. “Let’s keep going !”

With their confidence growing, the team pressed forward, ready for the next challenge inside the shadow pyramid.

To be continued…

Stay tuned for Chapter 4: The room of ratios

fundamentals of mathematics

Types of progression (AP,GP,HP progression)

Published

2 months ago

February 15, 2025

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Types of progression

A progression is a sequence of numbers following a specific pattern. The most common types of progression are:

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference (d).

1. General form of an AP: An arithmetic sequence is written as

a, a+d, a+2d, a+3d+…

Where

a=first term
d=common difference
n=number of terms
l=last term

2. The nth term of an AP is given by

3. Sum of first n terms

The sum of the first n terms is

Or,

Example 1: Find the 10th term of the AP: 3, 7, 11, 15,…

Here,

a=3, d=7-3=4, n=10

Using:

So, the 10th term is 39.

Example 2: Find the sum of the first 20 terms of the AP: 5, 8, 11, 14,…

Here,

a=5,d=8-5=3,n=20

Using

So the sum of the first 20 terms is 670.

Geometric Progression

A geometric progression is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed constant called the common ratio (r).

General form of a GP