Question

In Exercises \(19 - 28 ,\) determine the limit graphically. Confirm algebraically. $$\lim _ { x \rightarrow 0 } \frac { x + \sin x } { x }$$

Short answer

The limit as \(x \rightarrow 0\) for the function \(\frac{x + \sin x}{x}\) is equal to 2.

Step-by-step solution

Identify the Indeterminate Form

Rewrite the function and see if the expression is an indeterminate form by substituting \(x=0\) in the equation \(\frac{x + \sin x}{x}\). If the result is an indeterminate form, then use L'Hopital's rule to simplify the expression.

Apply L'Hopital's Rule

L'Hopital's rule states that in the case of indeterminate forms, you can differentiate the numerator and the denominator of the function and then take the limit. This gives the same result as if you took the limit of the function first. Differentiate the function \(x + \sin x\) and \(x\) which give as \(1 + \cos x\) and \(1\) respectively.

Take the Limit

Substitute \(x=0\) in your new derived function \(\frac{1 + \cos x}{1}\) and find the new limit.

Key concepts

L'Hopital's Rule

Understanding L'Hopital's Rule is pivotal when tackling calculus related to limits, especially when you encounter indeterminate forms. This rule provides a method for finding the limit of a quotient where both the numerator and the denominator approach zero or infinity. In simpler terms, it converts a tricky limit problem into a simpler one by allowing differentiation of the top and bottom parts separately.

L'Hopital's Rule applies under the specific condition that an indeterminate form of type 0/0 or ∞/∞ is present after substitution. If this condition is met, the rule instructs us to differentiate the numerator and the denominator individually and then to take the limit again. An essential requirement for L'Hopital's Rule is that these differentiations can be performed. The process can be repeated as necessary, applying the differentiation and limit-taking each time, until a determinate form or an easily calculable limit emerges.

The underlying mathematical beauty of L'Hopital's Rule lies in its reliance on the idea that the ratio of the rates of change (gradients) of the numerator and denominator as they approach a certain point may give insight into the behavior of the original ratio at that point. This is incredibly useful in understanding the behavior of functions without the need for complex algebraic manipulations.

Indeterminate Forms

When evaluating limits in calculus, indeterminate forms can be particularly troublesome. These are expressions obtained during a limit process where the limit does not point to a specific value but essentially seems to be stuck between possibilities, like 0/0, ∞/∞, 0∙∞, ∞ - ∞, 1^∞, 0^0, and ∞^0.

Indeterminate forms don't give us enough information on their own to decide the limit's value. The expression 0/0, for instance, could represent anything from -∞ to ∞ or any real number in between because depending on the exact behavior of the numerator and denominator as they approach zero, the effects can cancel out or amplify in unpredictable ways. The existence of indeterminate forms is a strong hint that a deeper analysis of the function is required, often involving techniques like L'Hopital's Rule, algebraic manipulation, factoring, or conjugation.

Being able to recognize indeterminate forms is critical because it signals when special techniques need to be employed in order to properly evaluate the limit.

Differentiation

Differentiation is one of the fundamental concepts of calculus. It refers to the process of calculating the derivative, which is the rate at which a function is changing at any point. For a function y=f(x), the derivative f'(x) at a point x measures how fast the value of y is changing per unit change in x.

It's important to understand that differentiation gives us a powerful tool for analyzing functions beyond simply computing slopes. It allows us to determine the maximum and minimum points, points of inflection, and concavity of graphs, as well as solve problems involving motion, optimization, and rates of change in various contexts. The derivation process involves rules and formulas such as the power rule, product rule, quotient rule, and chain rule.

In the context of limit problems, particularly those involving indeterminate forms, differentiation comes to the rescue through techniques like L'Hopital's Rule. By differentiating the numerator and denominator separately, it's often possible to resolve the ambiguity of the indeterminate form and find the precise value of the limit.