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L’Hospital’s Rule in Calculus

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L’Hospital’s Rule in Calculus

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Introduction:

The L'Hospital's rule is the most famous mathematical tool among students because of its effectiveness and simplicity. The L’Hospital’s rule is a general approach for evaluate the indeterminate forms like 0/0 or ∞/∞. L'Hospital's rule is applied to calculate the limits of indefinite forms for calculus derivatives. It is possible to use the L Hospital rule many times.

This rule can be used again and again since it 0/0 and form does not vanish. The L'- Hospital's Rule (L'Hôpital's rule) is the most fundamental rule in calculus. This rule analyzes the limits that involve the undetermined forms by using derivatives. In this article we will discuss the statement and proof of L’Hospital’s Rule, its examples, applications and frequently asked questions.

Statement of L’Hospital’s Rule:

Let v′ (x) and w′ (x) exist and w′ (x) is not equal to zero for all value of x in an interval (a, b]. If Lim x→ +m (v(x)) = If Lim x→ +m (w(x)) = 0 and Lim x→ +m (v′ (x) / w′ (x)) exist, then Lim x→ +m (v (x) / w (x)) / Lim x→ +m (v′ (x) / w′ (x)).

Proof:

We suppose that f (m) = w (m) = 0 as we know that if limit is applied on the constant function its value is remains same. Since w (x) = 0 otherwise w′ (x) = 0 at some x which belongs to open interval (a, b).

If we define the function z (x) = v (x) – (v (n) / w (n)). V(x), then z (m) = 0 = z (n), moreover z is continuous on the closed interval [a, b] and if define the function z in form of derivative as, z′ (x) = v′ (x) - (v (n) / w (n)). w′ (x) which is exist on the open interval (a, b).

According to Rolle’s Theorem there exist some x which belongs to open interval (a, b) such that z′ (x) = 0.

Hence,

v′ (x) / w′ (x) – v (n) / w (n)

As we know that a < x < b, it gives.

Lim x→ +m (v (x) / w (x)) = Lim x→ +m (v′ (x) / w′ (x))

Examples:

Example No.1:

Given that:

Lim x→ 0 (2x / x) =.... (1)

Solution:

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By apply the limit on 2x / x, according to quotient rule of limit,

Lim x→ 0 (2x / x) = Lim x→ 0 (2x) / Lim x→ 0 (x)

According to constant multiple rule of limit on the factor Lim x→ 0 (2x),

Lim x→ 0 (2x / x) = 2. Lim x→ 0 (x) / 0

Lim x→ 0 (2x / x) = 2. (0) / 0

Lim x→ 0 (2x / x) = 0/ 0 (Indeterminate Form)

Now apply the L’Hospital’s Rule (differentiate the form) on equation no.1 It gives,

Lim x→ 0 (2x / x) = Lim x→ 0 (2 / 1)

Now apply the limit,

2 / 1 = 2

‘Hence

Lim x→ 0 (2x / x) = 2

Example No.2:

Given that:

Lim x→ 0 (sin(x) / x) =.... (1)

Solution:

By apply the limit on (sin(x) / x), according to quotient rule of limit,

Lim x→ 0 (sin(x) / x) = Lim x→ 0 (sin(x) / Lim x→ 0 (x)

Lim x→ 0 (sin(x) / x) = 0/ 0 Indeterminate Form)

Now apply the L’Hospital’s Rule (differentiate the form) on equation no.1 It gives,

Lim x→ 0 (sin(x) / x) = Lim x→ 0 (-Cos(x) / 1)

Now apply the limit,

- Cos (0) / 1 = -1 / 1 = -1

‘Hence

Lim x→ 0 (sin(x) / x) = -1


Example No.3:

Given that:

Lim x→ ∞ (x4 / 2).... (1)

Solution:

By apply the limit on (x4 / 2x), according to quotient rule of limit,

Lim x→∞ (x4 / 2x) = Lim x→∞ (x4) / Lim x→∞ (2x)

According to constant multiple rule of limit on the factor Lim x→∞ (2x),

Lim x→∞ (x4 / 2x) = ∞/ ∞ (Indeterminate Form)

Now apply the L’Hospital’s Rule (differentiate the form) on equation no.1 It gives,

Lim x→∞ (x4 / 2x) = Lim x→∞ (4 x3 /2) ..... (2)

Apply the limit it gives ∞/ ∞ which is still intermediate form.

Now again apply the L ‘Hospital Rule on equation no.2 it gives,

⇒ Lim x→∞ 12x / 0

⇒ ∞

Hence,

Lim x→ ∞ (x4 / 2x) = ∞ which is undefined here Indeterminate form is vanished.

Applications of L’Hospital’s RULE:

For evaluating Indeterminate forms like 0/0 or ∞/∞, use the L’Hospital’s rule. L'Hospital's theorem is used in calculus to evaluate the limits of derivatives with indeterminate forms. This rule can be applied numerous times. This rule retains an indefinite form after each application, even if we only use it once.


FAQs:

1. What are Indeterminate forms in mathematics?

Two functions are involved in an indeterminate form, whose limit cannot be derived directly from the limits of the individual functions.

There is different Indeterminate forms like,

  • 0 / 0
  • ∞ - ∞
  • 0 ^∞
  • 1 ^∞

2. Why this rule is called the L 'Hospital Rule?

This rule of calculus is called L 'Hospital Rule because it carries the name of the Swiss mathematician Johann Bernoulli, who gave the formula to the French mathematician Guillaume-François-Antoine, marquis de L'Hospital, who was his student.

2. When L 'Hospital Rule is applicable?

For evaluating indeterminate forms like 0/0 or ∞/∞, use the L’Hospital’s rule. L'Hospital's theorem is used in calculus to evaluate the limits of derivatives with indeterminate forms. This rule can be applied many times until Indeterminate form is not vanished.

4. Give one example of L 'Hospital Rule?

Given that:

Lim x→ 0 (2x / x) =..... (1)

Solution:

By apply the limit on 2x / x, according to quotient rule of limit,

Lim x→ 0 (2x / x) = Lim x→ 0 (2x) / Lim x→ 0 (x)

According to constant multiple rule of limit on the factor Lim x→ 0 (2x),

Lim x→ 0 (2x / x) = 2. Lim x→ 0 (x) / 0

Lim x→ 0 (2x / x) = 2. (0) / 0

Lim x→ 0 (2x / x) = 0/ 0 (Indeterminate Form)

Now apply the L’Hospital’s Rule (differentiate the form) on equation no.1 It gives,

Lim x→ 0 (2x / x) = Lim x→ 0 (2 / 1)

Now apply the limit,

2 / 1 = 2

‘Hence

Lim x→ 0 (2x / x) = 2

5. What is the advantage of L’Hospital’s Rule?

The L'Hospital's rule is the most famous mathematical tool among students because of its effectiveness and simplicity.



Related Links:


  • Limit of a Function in Mathematics.
    Limits plays an important part in calculus as well as in our daily life. The concept of continuity, derivative and integral is based on limits. It is a most basic concept of calculus.
  • concept of Infinity in Mathematics
    Infinity is most important term which is used in every field of life as well as every field of science and arts. Everything which is unbounded refer to infinity.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2022 Kinza Javaid

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