Question

Evaluate limit and justify your answer. $$\lim _{x \rightarrow 1}\left(\frac{x+5}{x+2}\right)^{4}$$

Short answer

Function: $$\lim_{x \rightarrow 1}\left(\frac{x+5}{x+2}\right)^{4}$$ Answer: The limit of the given function as x approaches 1 is 16.

Step-by-step solution

Evaluate the limit of the rational function

Find the limit of the rational function as x approaches 1: $$\lim_{x \rightarrow 1}\left(\frac{x+5}{x+2}\right)$$ As it is a rational function, and the denominator is not zero when x=1, simply substitute x=1 to find the limit: $$\lim_{x \rightarrow 1}\left(\frac{x+5}{x+2}\right) = \frac{1+5}{1+2} = \frac{6}{3} = 2$$

Use the power rule to find the limit of the entire function

The limit of the rational function has been found to be 2. Now, use the power rule to find the limit of the entire function: $$\lim_{x \rightarrow 1}\left(\frac{x+5}{x+2}\right)^{4} = \left(\lim_{x \rightarrow 1}\left(\frac{x+5}{x+2}\right)\right)^{4}$$ We found in step 1 that the limit of the rational function as x approaches 1 is 2, so now substitute this into the power rule: $$\lim_{x \rightarrow 1}\left(\frac{x+5}{x+2}\right)^{4} = 2^{4} = 16$$

Write out the final answer

The limit of the given function as x approaches 1 is 16. So we have: $$\lim_{x \rightarrow 1}\left(\frac{x+5}{x+2}\right)^{4} = 16$$

Key concepts

Limits in Calculus

In calculus, limits are fundamental to understanding how functions behave as they approach a specific point. A limit allows us to examine the value that a function 'approaches' as the input (or independent variable) gets closer to some number. Imagine walking towards a door; the closer you get, the more you can tell about the door's color, even if you don't actually touch it. Similarly, by evaluating a limit, we observe a function's behavior near a particular point without necessarily reaching that point.

Calculating the limit of a function like \( \lim_{{x \to a}} f(x) \) involves figuring out what value the function \(f(x)\) is heading towards as \(x\) approaches \(a\). If \(f(x)\) approaches a single number as \(x\) gets infinitely close to \(a\), that number is the limit of the function at \(a\). If the function does not approach a single value, we say the limit does not exist or is infinite. Limits can be evaluated through direct substitution if the function is continuous at that point, or by applying various techniques like factoring, rationalizing, or using L'Hôpital's rule when dealing with more complex situations.

Rational Functions

Rational functions are ratios of two polynomials. They are expressed in the form \(f(x) = \frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomial functions and \(q(x)\) is not equal to zero. These functions can have points where they are undefined — specifically, wherever the denominator \(q(x)\) equals zero. These points are known as 'discontinuities' or 'singularities.'

Rational functions can exhibit various behaviors like vertical and horizontal asymptotes, which express their behavior at infinite distances, or near undefined points. For instance, as the value of \(x\) gets very large or small, the function may approach a specific value, known as a horizontal asymptote, or may increase or decrease without bound, indicative of a vertical asymptote. Learning how to handle rational functions includes understanding how to simplify them, find their domains, and importantly, how to evaluate their limits.

Power Rule for Limits

The power rule for limits is a powerful tool (pun intended) for quickly finding the limit of a function raised to a power. Formally, if we have a limit \( \lim_{{x \to a}} f(x)^n \) where \( f(x) \) approaches a limit \( L \) and \( n \) is a positive integer, then the limit of the function raised to the power of \( n \) is simply \( L^n \).

For example, once we establish that \( \lim_{{x \to a}} f(x) = L \) through direct substitution or other methods, applying the power rule is straightforward—just raise the limit \(L\) to the power \(n\): \( \lim_{{x \to a}} f(x)^n = L^n \). This rule saves time and simplifies the calculation process, making it easier to focus on the specific behavior of functions as they approach certain values, without getting bogged down by algebraic complexities.

Limit Evaluation

Evaluating limits is not always as simple as plugging in numbers; it often requires more insight into the function's form. Consider the limit \( \lim_{{x \to a}} f(x) \). If \(f(x)\) is continuous at \(x = a\), then direct substitution can be used. However, if direct substitution results in an indeterminate form like \(0/0\) or \(\infty/\infty\), then other techniques, such as factoring, rationalizing, or applying L'Hôpital's rule, may be necessary.

For rational functions like in the given exercise, if substituting \(x = a\) into both the numerator and denominator results in non-zero values, then the function is continuous at that point, and direct substitution yields the correct limit. This approach was used in the solution to find that \( \lim_{{x \to 1}} (\frac{x+5}{x+2})^4 = 16 \) after confirming that the function was well-behaved at \(x = 1\). Understanding how to evaluate limits accurately is vital because limits are foundational to the entire structure of calculus, particularly in defining derivatives and integrals.