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Glossary

D

Denominator

Criticality: 1

The expression or function located below the division bar in a fraction or rational expression.

Example:

In the limit lim⁡x→∞3x2−87x2+21\lim_{x \to \infty} \frac{3x^2-8}{7x^2+21}, 7x2+217x^2+21 is the denominator.

Derivatives

Criticality: 3

The instantaneous rate of change of a function, representing the slope of the tangent line to the function's graph at a given point, found using specific differentiation rules.

Example:

To apply L'Hôpital's Rule, you must correctly find the derivatives of both the numerator and the denominator, such as knowing that the derivative of tan⁡(x)\tan(x) is sec⁡2(x)\sec^2(x).

I

Indeterminate Forms

Criticality: 3

Expressions that arise when directly substituting a value into a limit, such as 0/0 or ±∞/∞, which do not immediately reveal the limit's true value and require further analysis.

Example:

When trying to find lim⁡x→2x2−4x−2\lim_{x \to 2} \frac{x^2-4}{x-2}, direct substitution results in 00\frac{0}{0}, which is an indeterminate form.

L

L'Hôpital's Rule

Criticality: 3

A powerful theorem used to evaluate indeterminate limits of the form 0/0 or ±∞/∞ by taking the derivatives of the numerator and denominator separately.

Example:

To evaluate lim⁡x→0ex−1x\lim_{x \to 0} \frac{e^x - 1}{x}, we can apply L'Hôpital's Rule to get lim⁡x→0ex1=1\lim_{x \to 0} \frac{e^x}{1} = 1.

Limits

Criticality: 3

The value that a function approaches as its input approaches a certain value, a foundational concept in calculus for understanding function behavior.

Example:

The entire purpose of L'Hôpital's Rule is to evaluate challenging limits that initially appear indeterminate.

N

Numerator

Criticality: 1

The expression or function located above the division bar in a fraction or rational expression.

Example:

In the limit lim⁡x→∞3x2−87x2+21\lim_{x \to \infty} \frac{3x^2-8}{7x^2+21}, 3x2−83x^2-8 is the numerator.

Q

Quotient Rule

Criticality: 2

A fundamental differentiation rule used to find the derivative of a function that is the ratio of two differentiable functions, distinct from L'Hôpital's Rule.

Example:

If you need to find the derivative of f(x)=ln⁡(x)x3f(x) = \frac{\ln(x)}{x^3}, you would use the Quotient Rule, not L'Hôpital's Rule, as it's a derivative problem, not a limit evaluation.