Question

find the limit $$ \lim _{t \rightarrow 4} \frac{t+4}{t^{2}-16} $$

Short answer

The limit of the given function as \( t \) approaches 4 is undefined.

Step-by-step solution

Factorize the denominator

The denominator \( t^{2}-16 \) is a difference of two squares and can be factorized into \( (t-4)(t+4) \). Therefore, the function can be re-written as \( \frac{t+4}{(t-4)(t+4)} \).

Simplify the expression

Now, cancel out the common factor \( t+4 \) from the numerator and the denominator. The function can be then written as \( \frac{1}{t-4} \).

Substitute the limit value

Once the function has been simplified, substitute \( t = 4 \) into the function. The limit is then \( \lim_{t\rightarrow 4} \frac{1}{t-4} = \frac{1}{4-4} = \frac{1}{0} \). Since we cannot divide by zero, we come to the conclusion that the limit is undefined.

Key concepts

Factorization

Factorization is a mathematical process where we express a number or a polynomial as a product of its factors. This is a crucial step in solving limit problems, especially when dealing with complex expressions.

In the context of calculus limits, factorization helps simplify expressions, making it easier to evaluate the limit. In our exercise, we apply factorization to the denominator \( t^2 - 16 \). This expression is a perfect example of a 'difference of squares'. This is a common algebraic technique where an equation follows the form \( a^2 - b^2 \) and can be rewritten as \((a-b)(a+b)\).

For our exercise, \( t^2 - 16 \) is factored into \((t-4)(t+4)\). Simplifying expressions through factorization is a helpful tool to cancel out terms that might complicate finding the limit, just as with the \( t+4 \) term in our problem. Thus, factorization not only makes the expression simpler but also avoids undefined expressions further in calculations.

Undefined Limits

In calculus, understanding the concept of undefined limits is fundamental. An undefined limit occurs when, as a function approaches a certain point, it does not yield a particular number. This usually involves dividing by zero or other circumstances where the function's behavior doesn't settle toward a specific value.

In our exercise involving the limit \( \lim_{t \rightarrow 4} \frac{1}{t-4} \), substituting \( t = 4 \) makes the denominator zero, leading to division by zero, which is undefined. When a limit yields an undefined result, this signifies that the limit does not exist in the usual sense.

• If the numerator approaches a non-zero value and the denominator approaches zero, the limit is typically deemed undefined.
• A deeper analysis might show a tendency towards infinity, but it is formally labeled undefined without further context.

Thus, understanding and identifying undefined limits is essential to comprehending the behavior of functions at specific points.

Difference of Squares

The difference of squares is a vital algebraic identity used frequently in simplifying polynomial expressions, especially in calculus problems involving limits. This concept revolves around the expression \( a^2 - b^2 \) which can always be factored into \( (a-b)(a+b) \).

Recognizing this pattern allows you to break down complex polynomials, facilitating further simplification or even allowing for cancelation of certain terms. In contexts like our example, \( t^2 - 16 \), seeing it as a difference of squares \((t^2 - 4^2)\) allows us to quickly apply the identity to get \((t-4)(t+4)\).

Mastering the difference of squares identity can aid a student significantly, not only in solving limits but also in other areas of algebra that require polynomial simplification. It condenses the problem, helping isolate and resolve terms much more efficiently.