Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 0} x \cos x$$

Short answer

Answer: The limit of the function \(x\cos x\) as \(x\) approaches 0 is 0.

Step-by-step solution

State the limit problem

We would like to find the limit of the function \(x\cos x\) as \(x\) approaches 0. So, we want to find the value of: $$\lim _{x \rightarrow 0} x \cos x$$

Applying limit laws (product rule)

Remind that the limit of the product is equal to the product of the limits, if both limits exist. We can apply the product rule to find the limit: $$\lim _{x \rightarrow 0} x \cos x = \left(\lim _{x \rightarrow 0} x\right) \left(\lim _{x \rightarrow 0} \cos x\right)$$

Find the limit of \(x\) as \(x\) approaches 0

As x approaches 0, the limit of \(x\) is 0, that is: $$\lim _{x \rightarrow 0} x = 0$$

Find the limit of \(\cos x\) as \(x\) approaches 0

As x approaches 0, the limit of \(\cos x\) is \(\cos 0=1\), that is: $$\lim _{x \rightarrow 0} \cos x = 1$$

Compute the limit of the product

Now that we have both limits, we can compute the limit of the product by multiplying the two limits we found in steps 3 and 4: $$\lim _{x \rightarrow 0} x \cos x = (0)(1) = 0$$ The limit of the function \(x\cos x\) as \(x\) approaches 0 is equal to 0.

Key concepts

Limit Laws

Limit laws are fundamental tools in calculus that help simplify the process of finding limits of functions. These rules allow us to break down complex expressions into simpler parts, making it easier to compute a limit. A few essential limit laws include the sum law, product law, and quotient law.
- The **sum law** states that the limit of the sum of two functions is the sum of their limits, provided both individual limits exist.
- The **product law** is particularly important and states that the limit of the product of two functions is the product of their respective limits, again assuming both limits exist.
- The **quotient law** gives that the limit of a quotient of two functions is the quotient of their limits, as long as the limit of the denominator is not zero.
In this exercise, the product law is used to find the limit of the expression \(x\cos x\) by considering the limits of \(x\) and \(\cos x\) separately. This method greatly simplifies complex limit problems and is one of the cornerstones of learning calculus.

Product Rule

The product rule in limit calculations is an essential principle that allows us to find the limit of a product by taking the product of the limits of individual functions. This is extremely helpful when dealing with expressions where multiplication is involved.
To apply the product rule correctly, ensure that each limit you need to compute exists. In our example, we look at \(\lim_{x \rightarrow 0} x\) and \(\lim_{x \rightarrow 0} \cos x\).
- It's clear that \(\lim_{x \rightarrow 0} x = 0\) because as \(x\) approaches zero, the value of \(x\) itself goes to zero.
- For \(\lim_{x \rightarrow 0} \cos x = 1\), since cosine is a continuous trigonometric function, its value at zero is well-defined and equals one.
Using the product rule, we then multiply the resulting limits: \(0 \times 1 = 0\). Thus, the limit \(\lim_{x \rightarrow 0} x\cos x\) becomes 0. This straightforward approach using the product rule can solve many problems efficiently.

Trigonometric Limits

Trigonometric limits involve evaluating limits where trigonometric functions are present. These are common scenarios, especially in calculus, because functions like sine, cosine, and tangent appear frequently in problems.
Understanding trigonometric limits is crucial since these functions exhibit unique behaviors around certain points, such as angles where they reach maxima or minima.
The most basic trigonometric limits to note include:
- \(\lim_{x \rightarrow 0} \sin x / x = 1\): This limit often appears when studying oscillatory motion or wave functions.
- \(\lim_{x \rightarrow 0} \cos x = 1\): Similar to our example problem, cosine approaches 1 as \(x\) approaches zero because cosine at zero equals 1.
In the example given, \(\cos x\)'s limit as \(x\) approaches 0 contributes to solving the problem along with the limit of \(x\), demonstrating how trigonometric and algebraic concepts often intertwine in calculus problems.