Question

Use analytical methods to evaluate the following limits. $$\lim _{n \rightarrow \infty}\left(n \cot \frac{1}{n}-n^{2}\right)$$

Short answer

Question: Determine the limit of the following expression as n approaches infinity: \(\lim _{n \rightarrow \infty}\left(n \cot \frac{1}{n}-n^{2}\right)\) Answer: The limit of the given expression as n approaches infinity is 0.

Step-by-step solution

Rewrite the expression using trigonometric identity

Rewrite the given expression using the identity \(\cot(x) = \dfrac{1}{\tan(x)}\): $$\lim _{n \rightarrow \infty}\left(n \cot \frac{1}{n}-n^{2}\right)= \lim _{n \rightarrow \infty}\left(n \cdot \dfrac{1}{\tan(\frac{1}{n})}-n^{2}\right)$$

Simplify the expression

Simplify the expression: $$\lim _{n \rightarrow \infty}\left(n \cdot \dfrac{1}{\tan(\frac{1}{n})}-n^{2}\right) = \lim _{n \rightarrow \infty}\left(\dfrac{n}{\tan(\frac{1}{n})}-n^{2}\right)$$ Now, if we try to plug in \(n \rightarrow \infty\) directly, we'll get an indeterminate form of \(\dfrac{\infty}{0}\) which means we need to use L'Hopital's rule to evaluate this limit.

Apply L'Hopital's rule with substitution

Let \(x = \frac{1}{n}\). Then, as \(n \rightarrow \infty\), \(x \rightarrow 0\). The given expression becomes: $$\lim_{x \rightarrow 0}\left(\dfrac{\dfrac{1}{x} }{\tan(x)}-\dfrac{1}{x^2}\right)$$ To apply L'Hopital's rule, we need to differentiate both numerator and denominator with respect to x: Numerator: $$\frac{d}{dx}\left(\dfrac{1}{x} - \dfrac{1}{x^2}\right) = - \dfrac{1}{x^2} + \dfrac{2}{x^3}$$ Denominator: $$\frac{d}{dx}\left(\tan(x)\right) = \sec^2(x)$$

Apply L'Hopital's rule and simplify the expression

Now we can apply L'Hopital's rule to find the limit: $$\lim_{x \rightarrow 0}\left(\dfrac{- \dfrac{1}{x^2} + \dfrac{2}{x^3}}{\sec^2(x)}\right)$$ Plug in \(x = 0\): $$\dfrac{- \dfrac{1}{0^2} + \dfrac{2}{0^3}}{\sec^2(0)} = \dfrac{0}{1} = 0$$ Therefore, the limit of the given expression is 0.

Key concepts

L'Hopital's Rule

One of the most powerful tools in calculus for dealing with indeterminate forms is L'Hôpital's rule. This rule allows us to compute limits which initially present an indeterminate form by differentiating the numerator and denominator separately and then taking the limit of the resulting expression. Indeterminate forms may include 0/0 or ∞/∞, which do not produce immediate answers and require further manipulation to evaluate.

L'Hôpital's rule can be invoked when we see these indeterminate forms, but it's important to understand that before using the rule, we should try to simplify the expression as much as possible. If simplification does not resolve the indeterminacy, we can proceed with differentiation as specified by the rule. L'Hôpital's rule is often used in conjunction with a substitution if the limit is not readily apparent, as it was in the given exercise where a substitution x = 1/n was made.

Trigonometric Identities

Understanding trigonometric identities is crucial to solving calculus problems involving trigonometric functions. These identities are essentially equations that are true for all values of the variables involved. In calculus, and especially in limit problems, these identities provide a way to simplify complex expressions or convert them into forms that are easier to manage.

One key identity used in the given exercise is the cotangent identity cot(x) = 1/tan(x). By using this identity, an originally complicated looking limit can be transformed to an expression that is more amenable to the application of L'Hôpital's rule. Remember, knowledge of various trigonometric identities can save considerable time and effort in solving problems and is essential when your goal is to present a trigonometric function in a different form.

Indeterminate Forms

In calculus, indeterminate forms are expressions that do not have a definitive limit or value and, as a result, require additional steps to evaluate. The typical indeterminate forms encountered in limit problems include 0/0, ∞/∞, 0×∞, ∞-∞, 1^∞, and others. These forms arise because both the numerator and the denominator approach zero or infinity, and it's unclear how they will balance out.

In our example, when we directly substitute n → ∞ in the limit expression, we end up with the indeterminate form ∞/0. This does not provide us with a clear answer, and direct computation fails. When faced with an indeterminate form, it's often necessary to apply techniques such as L'Hôpital's rule, algebraic simplification, or trigonometric identities, depending on the problem at hand, to find a limit that can be determined.