Question

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 4^{+}} \frac{4-x}{x^{2}-16} $$

Short answer

The limit as x approaches 4 from the right of the function is -1/8.

Step-by-step solution

Rewrite using difference squares

Start by rewriting the denominator using the difference of squares formula, it becomes: \[\lim _{x \rightarrow 4^{+}} \frac{4-x}{(x-4)(x+4)}\]

Simplify the expression

Upon factoring out \(-1\) from the numerator, the expression becomes: \[\lim _{x \rightarrow 4^{+}} \frac{-1*(x-4)}{(x-4)(x+4)}\] This allows for the cancellation of the \((x-4)\) terms.

Find the limit

After the simplification in step 2, the limit becomes: \[\lim _{x \rightarrow 4^{+}} \frac{-1}{x+4}\] Substituting 4 into the 'x' in the denominator, it can be found: \[\frac{-1}{4+4} = -\frac{1}{8}\]

Key concepts

Difference of Squares

The difference of squares is a classic algebraic method that's often used to simplify expressions. It involves rewriting an expression where one variable is subtracted from another, both of which are squared. Mathematically, the difference of squares formula is expressed as:

• \(a^2 - b^2 = (a-b)(a+b)\)

When you encounter an expression like \(x^2 - 16\), you can view it as \((x)^2 - (4)^2\). According to the difference of squares concept, this turns into \((x-4)(x+4)\). Utilizing this approach simplifies the denominators in limits and other rational expressions. This factorization is particularly handy in calculus, especially when calculating limits, as it can help identify common factors that cancel out.

Simplifying Rational Expressions

Simplifying rational expressions is like simplifying fractions in arithmetic. You aim to make expressions simpler by canceling out terms. For example, if we have an expression like \(\frac{4-x}{(x-4)(x+4)}\), we notice that \(4-x\) can be rewritten by factoring out \(-1\) as \(-1(x-4)\). This step transforms our expression into:

• \(\frac{-1(x-4)}{(x-4)(x+4)}\)

Once rewritten, it's clear that the \((x-4)\) terms in the numerator and denominator can cancel each other. This simplifies the rational expression to \(\frac{-1}{x+4}\). By simplifying, we've often made the problem or calculation at hand much easier, paving the way for more straightforward limit calculations or evaluations.

One-sided Limits

One-sided limits are a specific type of limit in calculus where you only consider the behavior of a function as it approaches a particular point from one side: either the left or the right. In the original exercise, we are looking at the right-handed limit denoted as \(x \rightarrow 4^{+}\). This means we're approaching 4 from the right, or from values greater than 4.
When calculating these types of limits, the function's behavior can differ from one side to the other, which may affect whether the limit exists or what its value is.

• The calculation begins by simplifying the expression like any limit problem.
• Then, substitute the value the variable approaches, such as the point 4 in our exercise.

In our example, after simplification, we substitute 4 into \(\frac{-1}{x+4}\) to get \(-\frac{1}{8}\). Ensuring you're consistent with the direction of approach is crucial for correctly solving one-sided limits.