Question

Evaluate the given limit. $$ \lim _{t \rightarrow 3}\left\langle e^{t}, \frac{t^{2}-9}{t+3}\right\rangle $$

Short answer

The limit is \( \left\langle e^{3}, 0 \right\rangle \).

Step-by-step solution

Analyze Each Component of the Vector

The given vector is \( \left\langle e^{t}, \frac{t^{2}-9}{t+3} \right\rangle \). It consists of two components: \( e^{t} \) and \( \frac{t^{2}-9}{t+3} \). We need to evaluate the limit for each component separately as \( t \rightarrow 3 \).

Evaluate the Limit of the First Component

For the first component, the limit is straightforward. We need to find \( \lim_{t \rightarrow 3} e^{t} = e^{3} \). Since \( e^{t} \) is continuous, the limit is simply the value of the function at \( t = 3 \), which is \( e^{3} \).

Simplify the Second Component

For the second component, \( \frac{t^{2}-9}{t+3} \), notice that the numerator \( t^{2}-9 \) can be factored as \( (t-3)(t+3) \). Thus, \( \frac{t^{2}-9}{t+3} = \frac{(t-3)(t+3)}{t+3} \). Simplify this by canceling \( t+3 \), which gives \( t-3 \), for \( t eq -3 \).

Evaluate the Limit of the Second Component

After simplifying, we have \( t-3 \). To find the limit as \( t \rightarrow 3 \), simply substitute \( t = 3 \): \( \lim_{t \rightarrow 3} (t-3) = 3 - 3 = 0 \).

Combine Limits of Vector Components

Now combine the limits of each component. The limit of the vector is \( \left\langle e^{3}, 0 \right\rangle \), which combines the results from Steps 2 and 4.

Key concepts

Vector Functions

Vector functions are mathematical expressions that involve vectors. These vectors may have one or several components, which are themselves functions dependent on one or more variables, commonly denoted by time \( t \). A vector function can be written in the form \( \mathbf{r}(t) = \langle f(t), g(t), h(t), \ldots \rangle \). Each component, \( f(t), g(t), h(t), \ldots \), is a distinct function of \( t \). Understanding vector functions involves:

• Identifying each component function within the vector.
• Knowing how these components transform as \( t \) varies.

In the context of evaluating limits, vector functions require analyzing the limit of each component separately. For instance, the original exercise uses a two-dimensional vector function: \( \left\langle e^{t}, \frac{t^2 - 9}{t + 3} \right\rangle \). This vector function combines an exponential component and a rational component, making it essential to manage each part individually as \( t \) approaches a specific value.

Limit of Functions

The limit of a function is a fundamental concept in calculus that describes the behavior of functions as the input approaches a particular value. This is particularly significant for evaluating vector functions, where each component function may have a distinct limit as the variable \( t \) approaches a given point.To find the limit of a vector function, we:

• Examine and determine the limit of each component function individually.
• Combine these limits to find the overall limit of the vector function.

For example, in our exercise problem, we analyze:- For \( e^{t} \), the exponential function, calculations are straightforward as it is continuous and smooth, hence the limit is simply \( e^{3} \) when \( t \) approaches 3.- For \( \frac{t^2 - 9}{t + 3} \), a more nuanced approach is required. The expression is simplified by factoring and eliminating common terms to solve for \( t \rightarrow 3 \).Grasping the limit of each part is crucial to predict the behavior of vector functions at specific points, ensuring continuity and precise calculations.

Continuity in Calculus

Continuity is a property of a function which conveys that the function does not have any abrupt changes, breaks, or holes. A continuous function allows for easy determination of limits as it ensures the value at a point equals the limit.This makes functions predictable and calculations more reliable. There are key points to consider:

• A function \( f(x) \) is continuous at a point \( x = c \) if \( \lim_{x \to c} f(x) = f(c) \).
• Continuity on an interval means the function is continuous at every point within that interval.
• If each component is continuous, the vector function is continuous.

For \( e^t \), the exponential function is continuous across its entire domain. This characteristic allowed the direct substitution and easy evaluation of \( \lim_{t \to 3} e^t \).In contrast, the rational expression \( \frac{t^2-9}{t+3} \) initially presents a discontinuity at \( t = -3 \) but remains continuous for other values, including closer to 3 after simplification.Understanding the continuity of each component is vital to effectively evaluate the limit and ensure accurate outcomes.