Question

, find the indicated limit. In most cases, it will be wise to do some algebra first. $$ \lim _{t \rightarrow 2} \frac{\sqrt{(t+4)(t-2)^{4}}}{(3 t-6)^{2}} $$

Short answer

The limit is \( \frac{\sqrt{6}}{9} \).

Step-by-step solution

Simplify the Expression Inside the Limit

The given limit expression is \( \lim _{t \rightarrow 2} \frac{\sqrt{(t+4)(t-2)^{4}}}{(3t-6)^{2}} \). First, notice that \( 3t - 6 = 3(t-2) \), so \((3t-6)^2 = [3(t-2)]^2 = 9(t-2)^2 \). Also, the expression inside the square root can be rewritten as \( \sqrt{(t+4)(t-2)^4} = (t-2)^2\sqrt{t+4} \) because \((t-2)^4 = ((t-2)^2)^2 \).

Rewrite the Limit Using Simplified Expression

After simplification, the limit expression now becomes:\[ \lim _{t \rightarrow 2} \frac{(t-2)^2 \sqrt{t+4}}{9(t-2)^2} \].

Cancel Out Common Factors

Notice that \((t-2)^2\) appears in both the numerator and the denominator. Cancel \((t-2)^2\) from the numerator and denominator to simplify the expression to:\[ \lim _{t \rightarrow 2} \frac{\sqrt{t+4}}{9} \].

Evaluate the Limit by Direct Substitution

Substitute \( t = 2 \) directly into the simplified function:\( \lim _{t \rightarrow 2} \frac{\sqrt{t+4}}{9} = \frac{\sqrt{2+4}}{9} = \frac{\sqrt{6}}{9} \).

Final Result

The limit has been evaluated to \( \frac{\sqrt{6}}{9} \). Thus, the solution is complete.

Key concepts

Simplifying Expressions

Simplifying expressions before working with limits is fundamental in calculus. This process often involves factoring or breaking down the algebraic components of a given function.
In our exercise, the function was initially complex. Simplifying helps reveal inherent patterns or cancellations.

We began by examining the expression inside the square root: \( (t+4)(t-2)^4 \). By recognizing that \( (t-2)^4 \) can be expressed as \(((t-2)^2)^2 \), we can simplify the square root operation to \( (t-2)^2 \sqrt{t+4} \).
This transformation helps to clean up the function, making further manipulation possible. Simplification is essential because it reduces the complexity, thereby making other steps more straightforward.

Additionally, recognizing \(3t-6\) as \(3(t-2)\) further clarifies the denominator, expressing it as \(9(t-2)^2\), which is useful for upcoming steps like cancellations.

Direct Substitution

Direct substitution is the process of evaluating the limit by simply plugging in the value of the variable approaching the limit.
Once the expression has been simplified adequately, plugging directly becomes possible and non-problematic.

After simplifying the expression in our example, we reached a form, \( \frac{\sqrt{t+4}}{9} \). At this stage, we can substitute \( t = 2 \) easily because no more complications like undefined terms are found.
This step is straightforward yet crucial because it determines the numerical value of the limit. The procedure implies simply entering the approaching value into the more accessible, final form of the function.
It is often the last step after unraveling the function with algebraic manipulation and simplification, leading to an effortless calculation of the limit value: \( \frac{\sqrt{6}}{9} \).

Algebraic Manipulation

Algebraic manipulation refers to using mathematical techniques to rearrange or modify expressions for simplification or solution.
In limits, it's pivotal for clearing complexities such as factors, that would otherwise hinder direct evaluation.

During algebraic manipulation in our problem, common factors were identified and canceled. Specifically, \((t-2)^2\) was present in both the numerator and denominator. By canceling these, we are left with a much simpler function that maintains the original's limit properties.
This critical step showcases how manipulation can simplify complex expressions without altering fundamental properties. Additionally, we rewrote the original denominator \((3t-6)^2\) as \(9(t-2)^2\), using basic algebraic rules.

Such thoughtful rearrangement helps identify cancellations and reduce the expression to a form where other methods like direct substitution can be aptly applied. Mastering these skills leads to efficient limit evaluation in calculus.