Problem 21 - Contemporary Precalculus

Question

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 1} \sqrt{x^{3}+6 x^{2}+2 x+5}$$

Step-by-Step Solution

Answer

Answer: The limit of the given function as x approaches 1 is \(\sqrt{14}\).

Step 1: Check if the limit exists

Substitute x with the value it approaches (1) in the expression inside the square root: $$1^3 + 6 * 1^2 + 2 * 1 + 5 = 1 + 6 + 2 + 5 = 14$$ Since the value inside the square root is non-negative, the limit exists.

Step 2: Find the limit

Now, let's find the limit as x approaches to 1: $$\lim _{x \rightarrow 1} \sqrt{x^{3}+6 x^{2}+2 x+5} = \sqrt{1^3+6*1^2+2*1+5} = \sqrt{14}$$ The limit of the given function as x approaches 1 is \(\sqrt{14}\).

Key Concepts

Understanding Limits in Calculus

In calculus, when we speak about the limit of a function, we're trying to understand the behavior of the function as the input values approach a certain point. This concept is crucial because it lays the foundation for continuity, derivatives, and integrals—all fundamental in studying change and area under curves.

Limits help us define the exact value that a function tends to, even if it doesn't actually reach that value. If the limit exists and is finite, it means our function approaches a specific number as the input gets close to a given value, even if the function doesn't actually reach that number at the point.

For example, imagine a point on a graph that's missing or a vertical asymptote where a function soars towards infinity—limits describe the function's approach without getting halted by these breaks in the graph.

Dealing with the Square Root Limit

Why Does the Square Root Add Complexity?

The presence of a square root in a limit, such as in our given problem, adds an extra layer of complexity. A square root function only takes on real values for non-negative inputs. Therefore, when evaluating limits that involve square roots, it's imperative to ensure that the expression under the square root sign is non-negative in the limit.

Our primary concern is to avoid undefined behaviors (such as taking the square root of a negative number) and to cautiously approach the limit from both the left and the right. If the expression inside the square root is non-negative as we approach our input value, and it remains so on either side of that value, the limit will exist.

Evaluating Limits

Evaluating limits is the process of finding the value that a function is approaching as the input approaches a certain value. In our example, this is done by a direct substitution method where the value that x is approaching (which is 1 in this case) is substituted directly into the function.

If the function is continuous at the point we're interested in, then the limit is simply the value of the function at that point. If the direct substitution method results in an indeterminate form, like 0/0, then alternative strategies are employed, such as factoring, rationalizing, or applying L'Hôpital's rule.

Limit Exists: A Closer Look

When evaluating the limit of the square root function in our exercise, we first check if the expression within the square root is non-negative as x approaches 1, which it is. Then, we substitute x with the limiting value, and if the resulting expression yields a real number, as it did in this case with \( \sqrt{14} \), we've successfully evaluated the limit.

Through this process, we gain a powerful tool for understanding how functions behave and can tackle more complex problems with confidence.