Glossary

Denominator

Criticality: 1

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

Example:

In the limit $\lim_{x \to \infty} \frac{3x^2-8}{7x^2+21}$ , $7x^2+21$ 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 $\tan(x)$ is $\sec^2(x)$ .

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 $\lim_{x \to 2} \frac{x^2-4}{x-2}$ , direct substitution results in $\frac{0}{0}$ , which is an indeterminate form.

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 $\lim_{x \to 0} \frac{e^x - 1}{x}$ , we can apply L'Hôpital's Rule to get $\lim_{x \to 0} \frac{e^x}{1} = 1$ .

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.

Numerator

Criticality: 1

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

Example:

In the limit $\lim_{x \to \infty} \frac{3x^2-8}{7x^2+21}$ , $3x^2-8$ is the numerator.

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

Glossary

Denominator

Criticality: 1

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

Example:

In the limit $\lim_{x \to \infty} \frac{3x^2-8}{7x^2+21}$ , $7x^2+21$ 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 $\tan(x)$ is $\sec^2(x)$ .

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 $\lim_{x \to 2} \frac{x^2-4}{x-2}$ , direct substitution results in $\frac{0}{0}$ , which is an indeterminate form.

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 $\lim_{x \to 0} \frac{e^x - 1}{x}$ , we can apply L'Hôpital's Rule to get $\lim_{x \to 0} \frac{e^x}{1} = 1$ .

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.

Numerator

Criticality: 1

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

Example:

In the limit $\lim_{x \to \infty} \frac{3x^2-8}{7x^2+21}$ , $3x^2-8$ is the numerator.

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