Question

Find \(\lim _{x \rightarrow 0} \frac{\tan x}{x}\)

Short answer

Answer: The limit of the function \(\frac{\tan x}{x}\) as \(x\) approaches \(0\) is \(1\).

Step-by-step solution

Identify indeterminate form

As x approaches 0, both the numerator and the denominator approach 0, giving an indeterminate form \(\frac{0}{0}\). So, we need to apply L'Hopital's Rule.

Differentiate the numerator and the denominator

L'Hopital's Rule involves differentiating the numerator and the denominator separately. We need to find the derivatives of both \(\tan x\) and \(x\): \(\frac{d}{dx}(\tan x) = \sec^2 x\) and \(\frac{d}{dx}(x) = 1\).

Apply L'Hopital's Rule

Since we have the required derivatives from Step 2, we can now apply L'Hopital's Rule: \(\lim _{x \rightarrow 0} \frac{\tan x}{x} = \lim _{x \rightarrow 0} \frac{\sec^2 x}{1}\).

Find the limit

Now, we can plug in \(x=0\) into the simplified expression: \(\lim _{x \rightarrow 0} \frac{\sec^2 x}{1} = \frac{\sec^2(0)}{1}= \frac{1}{1}=1\) So, the limit of the given function as \(x\) approaches \(0\) is \(1\).

Key concepts

L'Hopital's Rule

Understanding L'Hopital's rule is crucial for solving many problems in calculus, particularly when dealing with tricky limits. This rule comes to the rescue when a direct substitution in the limit leads to an indeterminate form like \frac{0}{0}\text{ or }\frac{\text{\text{infinity}}}{\text{\text{infinity}}}\.

L'Hopital's Rule states that if the limit of a function \frac{f(x)}{g(x)}\ as \(x\) approaches a certain value results in an indeterminate form, then the limit of this function will be the same as the limit of the derivatives of the top and bottom functions. So, if \(\text{lim }_{x \rightarrow c} f(x)\) and \(\text{lim }_{x \rightarrow c} g(x)\) both approach 0 or infinity, we can instead evaluate \(\text{lim }_{x \rightarrow c} \frac{f'(x)}{g'(x)}\), where \(f'(x)\) and \(g'(x)\) represent the derivatives of \(f(x)\) and \(g(x)\), respectively.

To apply L'Hopital's Rule appropriately, it's important to ensure that the following conditions are met:

• \(f(x)\) and \(g(x)\) are both differentiable near the point \(c\), except possibly at \(c\) itself.
• The limit \(\text{lim }_{x \rightarrow c} \frac{f'(x)}{g'(x)}\) exists or is infinite.
• The original limit leads to an indeterminate form 0/0 or infinity/infinity.

Once these criteria are satisfied, you can confidently use L'Hopital's Rule to uncover the elusive limit.

Indeterminate Forms

In calculus, indeterminate forms are expressions whose limits cannot be determined solely from the limits of their individual parts. Common examples include \(0/0\), \(0 \times \text{\text{infinity}}\), \(\text{\text{infinity}} - \text{\text{infinity}}\), \(1^\text{\text{infinity}}\), \(\text{\text{infinity}}^0\), and \(0^0\). These expressions require special techniques to evaluate, such as algebraic manipulation, limits of functions within the expression, or tools like L'Hopital's Rule.

In the example of finding \(\text{lim }_{x \rightarrow 0} \frac{\tan x}{x}\), the expression initially results in the indeterminate form 0/0. Students confronted with this situation should recognize that traditional methods of finding limits by direct substitution won't work and will have to resort to alternative strategies. Identifying indeterminate forms accurately is essential for applying the correct methods to find the limits of functions.

AP Calculus AB

AP Calculus AB is an advanced placement course that delves into the fundamentals of calculus, including limits, derivatives, integrals, and the Fundamental Theorem of Calculus. It's equivalent to a first-semester college calculus course. The curriculum focuses on developing students' understanding of mathematical concepts, working with mathematical expressions, and applying techniques to solve various problems.

The topic of limits, especially techniques like L'Hopital's Rule, plays a significant part in the curriculum, as it helps students tackle a broad range of problems that involve determining the behavior of functions as they approach specific points or infinity. Mastery in these subjects is not only beneficial for high school academics but is also crucial for students intending to pursue STEM fields in higher education. The example provided, \(\text{lim }_{x \rightarrow 0} \frac{\tan x}{x}\), is the kind of challenging problem students might encounter in the AP Calculus AB exam.

Differentiation

Differentiation is a fundamental concept in calculus that deals with determining how a function changes at any given point. It's how we find a function's derivative, which represents the rate of change. The process of differentiation involves taking the derivative of a function, which can give us the slope of the tangent line to the graph of the function at any point.

For example, when applying L'Hopital's Rule to the limit problem \(\text{lim }_{x \rightarrow 0} \frac{\tan x}{x}\), differentiation helps us transition from dealing with the complicated trigonometric function \(\tan x\) to the simpler power function \(\text{sec}^2 x\). The ability to differentiate correctly is crucial to finding accurate solutions to these limit problems. Knowing the rules of differentiation, such as the product rule, quotient rule, chain rule, and the derivatives of basic functions, equips students with the tools needed to handle a wide range of calculus problems, not just in AP Calculus AB but in higher-level mathematics as well.