The result we will be exploring is known as l'Hôpital's...

The result we will be exploring is known as l'Hôpital's Rule. One version of it says that if $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}$ is indeterminate, but $f$ and $g$ have derivatives at $a$ with $g^{\prime}(a) \neq 0$, then $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{f^{\prime}(a)}{g^{\prime}(a)}$. a. By taking the appropriate derivatives, apply the above result to the quotient $\frac{\ln x}{x^2-1}$ at the point $a=1$. Also check the result graphically by plotting, on the same axes, the quotient of the functions $f$ and $g$ defined by $f(x)=\ln x$ and $g(x)=x^2-1$ along with the quotient of their derivatives (not the derivative of the quotient!) as called for by l'Hôpital's Rule. What should happen, according to l'Hôpital's Rule, in your graphs? Does it indeed happen? b. Do the same as above for the quotient of functions given in Problem 1d.

The result we will be exploring is known as l'Hôpital's Rule. One version of it says that if $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}$ is indeterminate, but $f$ and $g$ have derivatives at $a$ with $g^{\prime}(a) \neq 0$, then $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{f^{\prime}(a)}{g^{\prime}(a)}$.
a. By taking the appropriate derivatives, apply the above result to the quotient $\frac{\ln x}{x^2-1}$ at the point $a=1$. Also check the result graphically by plotting, on the same axes, the quotient of the functions $f$ and $g$ defined by $f(x)=\ln x$ and $g(x)=x^2-1$ along with the quotient of their derivatives (not the derivative of the quotient!) as called for by l'Hôpital's Rule. What should happen, according to l'Hôpital's Rule, in your graphs? Does it indeed happen?
b. Do the same as above for the quotient of functions given in Problem 1d.

Key Concepts

l'Hôpital's Rule

l'Hôpital's Rule is a method for evaluating limits of indeterminate forms, typically 0/0 or ?/?, by replacing the original limit with the limit of the quotient of the derivatives of the functions involved. This rule requires that the functions are differentiable in a neighborhood around the point of interest (except possibly at the point itself), and that the derivative of the denominator is non-zero.

Indeterminate Forms

Indeterminate forms, such as 0/0 or ?/?, occur when substituting the limit into the function yields an undefined expression. Recognizing these forms is crucial as they signal the need for additional techniques, like l'Hôpital's Rule, to properly evaluate the limit by transforming it into a determinate form.

Derivatives

Derivatives represent the instantaneous rate of change of a function with respect to its variable. In the context of limits and l'Hôpital's Rule, taking the derivative of the numerator and the denominator allows one to simplify and evaluate limits where the original functions produce an indeterminate form.

Application of l'Hôpital's Rule

The application of l'Hôpital's Rule involves computing the derivatives of the numerator and denominator separately and then taking the limit of their quotient. This process often simplifies the expression and resolves the indeterminate form, yielding a determinate limit.

Graphical Interpretation of Limits

Graphical interpretation involves plotting functions and their derived counterparts to visually examine their behavior near a point. In the context of l'Hôpital's Rule, comparing the graphs of the original quotient and the quotient of the derivatives can provide empirical confirmation of the analytical limit, demonstrating that they converge to the same value.

Continuity and Differentiability

For l'Hôpital's Rule to be applicable, the functions involved must be differentiable around the point of interest, and the denominator's derivative should not be zero at that point. This ensures that both the original functions and their derivatives behave well enough for the rule to yield a valid limit.

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