Question

Find \(\lim _{t \rightarrow 0} A(t)\) or state that the limit does not exist. $$ A(t)=\left[\begin{array}{cc} t e^{-t} & \tan t \\ t^{2}-2 & e^{\sin t} \end{array}\right] $$

Short answer

Question: Determine the limit of the given matrix A(t) as t approaches 0, if it exists: \[ A(t) = \left[\begin{array}{cc} te^{-t} & \tan{t}\\ t^2-2 & e^{\sin{t}} \end{array}\right]\] Answer: The limit of the given matrix A(t) as t approaches 0 is: \[ \lim_{t \rightarrow 0} A(t) =\left[\begin{array}{cc} 1 & 0\\ -2 & 1 \end{array}\right] \]

Step-by-step solution

Finding limits of matrix elements

To find the limit of A(t) as t approaches 0, let's find the limit of each element in the matrix:

Limit of \(te^{-t}\)

We will find the limit of this element using L'Hôpital's rule, which states that if a function is in indeterminate form, the limit will be equal to the limit of the ratio of its derivative: $$ \lim_{t\rightarrow0} \frac{t}{e^{t}} \frac{d}{dt}\left(\frac{t}{e^{t}}\right) = \lim_{t\rightarrow0} \frac{1- t}{e^t} = \frac{1 - 0}{e^0} = 1 $$

Limit of \(\tan{t}\)

The limit of \(\tan{t}\) is straightforward since it is continuous at t=0: $$ \lim_{t\rightarrow0} \tan{t} = \tan{0} = 0 $$

Limit of \(t^2-2\)

Similarly, the limit of \(t^2-2\) is simple as the function is continuous: $$ \lim_{t\rightarrow0} (t^2-2) = (0)^2 - 2 = -2 $$

Limit of \(e^{\sin{t}}\)

As e^x is always continuous and \(\sin\) is also continuous, their composition is continuous: $$ \lim_{t\rightarrow0} e^{\sin{t}} = e^{\sin{0}} = e^0 = 1 $$

Combine the limits into a matrix

Now that we have found the limits of each element of the matrix, we can combine them to form the limit of the matrix A(t): $$ \lim_{t \rightarrow 0} A(t) =\left[\begin{array}{cc} 1 & 0\\ -2 & 1 \end{array}\right] $$

Key concepts

L'Hôpital's Rule

Understanding L'Hôpital's rule is a cornerstone for calculus enthusiasts, especially when encountering the mysterious indeterminate forms like 0/0 or \(\infty/\infty\).

L'Hôpital's rule elegantly guides us through these murky mathematical waters. It states that if you have a limit in one of those indeterminate forms, the limit of the original functions can be found by taking the limit of their derivatives instead. The punch line? Both the numerator and the denominator must be differentiable, and the denominator can't be zero.

Applying L'Hôpital's Rule

With our eyes on the prize, let's consider a function, say \( f(t)/g(t) \), tip-toeing towards an indeterminate form as \( t \) creeps toward a certain value. If \( f'(t) \) and \( g'(t) \) exist near this value and \( g'(t) \) isn't zero, then we can refocus our limit-searching efforts on \( f'(t)/g'(t) \). This swap can simplify our quest or at least give us a new expression that might be friendlier for limit hunting.

Back to our problem, we used L’Hôpital’s rule to find the limit of \( te^{-t} \) as \( t \) approaches 0. We faced \( 0/0 \) - classic indeterminate form territory. By taking the derivatives and applying the rule, the fog lifted, and we found our limit to be 1.

Matrix Elements Limit

If we peek into the realm of matrices, we notice that they're essentially a collection of individual elements, each possibly bearing its own function of \( t \). To inspect the limit of a matrix as \( t \) approaches a certain value, like the tricksy 0 in our example, the game plan is to probe the limit of each of its elements separately.

Individual vs. Collective Limits

Imagine you're at a party, and each person (an element) is easing into an enlightening fact (a limit) as the night unwinds (as \( t \)) goes to 0). You'd mingle and learn each person's story separately, right? That's exactly how we approach matrices too. It simplifies the complexity - breaking down a seemingly large task into smaller, manageable bites. Each limit we find for an element is like an interesting tidbit from one of our party-goers.

Once we've gathered all these facts, we assemble them into a matrix, similar to gathering all the snippets of conversations into a memorable party summary. In the provided problem, after determining limits for each function in the matrix elements, we could combine them into the final matrix limit as \( t \) approaches 0.

Continuous Functions

Continuous functions are the bread and butter of calculus - smooth, predictable, and without any abrupt breaks or holes. They're the friends you can rely on to gently pass you the baton in a relay race without dropping it.

Think of them like a playlist where each song seamlessly transitions into the next - no pauses or skips. If a function is continuous at a point, you can trace its graph with a pencil without lifting it from the page.

Continuity and Limits

Here's the deal: if a function energetically sprints towards a point and it's continuous there, the function's value at that point and the limit as it approaches that point are one and the same. It's this predictability that allows us to swiftly say the limit of \( \tan(t) \) as \( t \) approaches 0 is simply \( \tan(0) \) - which is a well-behaved 0. Similarly, the exponential function \( e^{\sin(t)} \) maintains its poise as \( t \) approaches 0, allowing us to assert that its limit is \( e^{0} \) or a comforting 1. Continuous functions make analyzing limits a walk in the park, as seen in this exercise.