Problem 54 - Calculus: Early Transcendentals

Question

Evaluate limit. $$\lim _{t \rightarrow 2} \frac{t^{2}+5}{1+\sqrt{t^{2}+5}}$$

Step-by-Step Solution

Answer

Question: Find the limit of the following expression as t approaches 2: $$\lim_{t \rightarrow 2} \frac{t^{2}+5}{1+\sqrt{t^{2}+5}}$$ Answer: The limit as t approaches 2 is $$\frac{9}{4}$$.

Step 1: Substitution

First, let's try substitution. Replace t with 2 in the expression and see if it gives a valid result: $$\frac{(2)^{2}+5}{1+\sqrt{(2)^{2}+5}} = \frac{9}{1+\sqrt{9}} = \frac{9}{1+3} = \frac{9}{4}$$ Since we got a valid result, the limit exists and we have found it.

Step 2: State the Final Answer

The limit as t approaches 2 is: $$\lim_{t \rightarrow 2} \frac{t^{2}+5}{1+\sqrt{t^{2}+5}} = \frac{9}{4}$$

Key Concepts

Substitution Method

The substitution method simplifies evaluating limits by directly replacing the variable with a specific value. In this exercise, the goal is to evaluate the limit of the function as the variable approaches a particular point. To apply the substitution method, replace the variable in the equation with the given value. For instance, if you need to find the limit of \(\lim_{t\rightarrow 2}\frac{t^2 + 5}{1 + \sqrt{t^2 + 5}}\), you substitute \(t\) with \(2\).
This substitution helps to check if it results in a real and finite number without complications like undefined expressions or indeterminate forms. If the expression resolves neatly, then the limit is that resulting value.

• Step-by-Step: Plug in the variable's value directly into the formula.
• If the outcome is a real number, the limit exists and equals this number.

By successfully replacing \(t\) with \(2\), and calculating \(\frac{9}{4}\), we see substitution provided a clear solution for this limit.

Evaluating Limits

Evaluating limits involves determining the value that a function approaches as the variable approaches a certain point. Limits are foundational in calculus, helping us understand the behavior of mathematical functions near specific points.
When evaluating, it's crucial to consider potential pitfalls such as indeterminate forms. In such cases, additional algebraic manipulation or special techniques might be required. However, the simplest cases, like our example, can often be solved by direct substitution if the function remains continuous and well-defined at the point of interest.

• The idea is to determine a function's behavior as you get infinitely close to a certain input.
• If a direct substitution results in a valid expression, the limit equals this value without any complications.

Thus, exploring limits enhances comprehension of how functions behave near boundaries and is instrumental in calculus.

Mathematical Expressions

A mathematical expression represents any conceivable part of a mathematical statement, be it a number, a variable, an operation, or more complex combinations of these. Understanding how to manipulate and evaluate expressions is crucial for grasping higher-level math concepts, including evaluating limits. Mathematical expressions follow specific rules and properties, such as the order of operations (PEMDAS/BODMAS), which dictate how to process complex expressions correctly.
In our original exercise, the expression \(\frac{t^2 + 5}{1 + \sqrt{t^2 + 5}}\) includes a polynomial term, a radical, and a fraction. These components combine to define the function we're analyzing the limit of.

• Expressions can include constants, variables, arithmetic operations, and functions like roots.
• Proper simplification often requires thorough attention to mathematical rules and properties.

Thus, becoming proficient in reading and simplifying these expressions is key to unlocking success in mathematics, particularly in calculus topics like limits.