Question

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

Short answer

Answer: The value of the limit as x approaches 1 for the expression $(\frac{x+5}{x+2})^4$ is 16.

Step-by-step solution

Analyze the limit of the rational function

We need to analyze the limit of the rational function inside the parentheses as x approaches 1. $$\lim _{x \rightarrow 1}\frac{x+5}{x+2}$$

Substitute the value of x into the rational function

Replace x with 1 in the rational function to obtain: $$\frac{1+5}{1+2}$$

Simplify the expression

Simplify the expression as follows: $$\frac{6}{3} = 2$$

Evaluate the limit of the composition

Now that we found the limit of the rational function to be 2, we will substitute this value into the composition and evaluate the limit. $$\lim _{x \rightarrow 1}\left(\frac{x+5}{x+2}\right)^{4} = 2^4 $$

Simplify the expression

Simplify the expression to find the final result: $$2^4 = 16$$ The limit of the given composition as x approaches 1 is 16.

Key concepts

Rational Functions Limits

Understanding the behavior of rational functions as they approach a certain value is pivotal in calculus. A rational function is a ratio of two polynomials, expressed in the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) is not equal to zero. When we're dealing with limits of rational functions, we're interested in finding the value that the function approaches as the input \( x \) gets infinitely close to a certain number, which in the given exercise is 1.

When calculating the limit of a rational function as \( x \) approaches a particular value, if direct substitution of \( x \) into the function doesn't lead to an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) we can usually find the limit by simply substituting the value of \( x \) into the function. In the exercise, when \( x \) was substituted with 1, the function didn't show an indeterminate form, which made it straightforward to evaluate the limit through simple substitution.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus that involves finding the value a function approaches as the input nears a specific point. The limit of a function at a certain point can either be a real number, infinity, or it may not exist. There are various methods to evaluate limits, and one commonly used technique is substitution, which works effectively when the function is continuous at the point of interest.

In our exercise, the step-by-step solution required us to evaluate the limit of a composed function. After simplifying the inner rational function through substitution, it became clear that the entire expression converges towards 16 as \( x \) approaches 1. It is important to note that continuous functions, including polynomials and rational functions where the denominator isn't zero, allow for direct substitution to evaluate limits, as they are well-behaved and predictable around the points of interest.

Substitution Method in Limits

The substitution method is a primary technique used to determine the limit of functions, especially when the function is continuous at the point we're considering. It simply involves replacing the variable \( x \) with the value it is approaching in the limit expression. If this direct substitution yields a numerical result rather than an indeterminate form, then that result is the limit of the function at that point.

In the example provided, the substitution method is painlessly applied because when \( x \) is replaced with 1, the function \( \frac{x+5}{x+2} \) presents no indeterminate form; it cleanly evaluates to 2. Therefore, we determine the limit of the composition by raising 2 to the fourth power, which equals 16. This task exemplifies the substitution method's efficiency in straightforward scenarios where the function's behavior is predictable at the point of interest.