Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 4} \frac{x^{2}-16}{4-x}$$

Short answer

Answer: The limit of the given function as x approaches 4 is -8.

Step-by-step solution

Factor the numerator

Factor the numerator of the given function \((x^2 - 16)\) as a difference of squares: $$(x^2 - 16) = (x - 4)(x + 4)$$

Rewrite the expression

Rewrite the given function in this factored form: $$\frac{x^2 - 16}{4 - x} = \frac{(x - 4)(x + 4)}{4 - x}$$

Cancel common terms

In order to deal with the indeterminate \(4-x\) term in the denominator, let's factor out a \(-1\) from \((4 - x)\). Now we have: $$\frac{(x - 4)(x + 4)}{4 - x} = \frac{(x - 4)(x + 4)}{-1(x - 4)}$$ Now, notice the common term \((x - 4)\). Cancel it out: $$\frac{(x - 4)(x + 4)}{-1(x - 4)} = -\frac{(x + 4)}{1}$$

Find the limit

Now, as the function simplifies to \(-(x+4)\), we can find the limit as x approaches 4: $$\lim_{x \rightarrow 4} -(x + 4) = - (4 + 4) = -8$$ Therefore, the limit of the given function as x approaches 4 is -8.

Key concepts

limit evaluation

In calculus, evaluating limits involves finding the value that a function approaches as the input approaches a certain point. It gives us insight into the behavior of functions near specific points, even if the function does not exist at those points. The notation \( \lim_{x \rightarrow a} f(x) \) is used to denote the limit of \( f(x) \) as \( x \) approaches \( a \).

To evaluate limits, it's often necessary to simplify the expression first. This might involve factoring, rationalizing, or using specific limits laws. Approaching an indeterminate form, such as \( \frac{0}{0} \), requires rewriting the expression in a simpler form until the limit can be applied. This approach helps avoid direct substitution which can otherwise lead to undefined expressions.

By using algebraic manipulation, limits that initially seem undefined can often be evaluated effectively, revealing the intended behavior of the function as it approaches the target value.

difference of squares

The difference of squares is a mathematical identity that's very helpful in algebra and calculus. It is the expression of the form \( a^2 - b^2 \), and it can be factored into \( (a - b)(a + b) \). This identity is a powerful tool for simplifying expressions.

In limit problems, like the one given, recognizing a difference of squares allows us to factor complex numerators or denominators effectively. For the expression \( x^2 - 16 \), you can see it as \( (x)^2 - (4)^2 \).

Applying the factoring of the difference of squares results in \( (x - 4)(x + 4) \).

• This makes canceling terms like \( (x - 4) \) from the numerator and denominator possible, simplifying the limit evaluation process.

Thus, recognizing and using the difference of squares can simplify many algebraic expressions encountered in calculus.

factoring expressions

Factoring expressions is an algebraic technique used to rewrite expressions as a product of simpler terms. This process is crucial in calculus for simplifying complex expressions and solving equations. Factoring often helps us deal with polynomial expressions effectively.

In the context of evaluating limits, factoring helps transform an indeterminate form like \( \frac{0}{0} \) into a more manageable expression. A common scenario in limits involves functions that initially present undefined behavior where substitution fails.

For example, in the limit problem \( \lim_{x \rightarrow 4} \frac{x^2-16}{4-x} \), factoring the expression \( x^2 - 16 \) as \( (x - 4)(x + 4) \) allows us to rewrite the original expression. Then factoring out \((-1)\) from \( (4 - x) \) aligns the terms for cancellation. When \( x - 4 \) is canceled from both the numerator and the denominator, the expression becomes defined.

• This simplification step is key in evaluating the limit successfully, transforming potential undefined behavior into a clear result.

Through practice, factoring becomes an intuitive approach to breaking down complex equations, leading to clearer solutions in calculus problems.