Evaluate The Following Limit:$\lim _{t \rightarrow 11} \frac{t-11}{\sqrt{t+5}-4}$

Feb 27, 2025 by ADMIN 82 views

**Evaluating Limits: A Step-by-Step Guide**

Introduction

Limits are a fundamental concept in calculus, and evaluating them is a crucial skill for any mathematician. In this article, we will evaluate the limit of the expression $\lim _{t \rightarrow 11} \frac{t-11}{\sqrt{t+5}-4}$ , which appears to be a challenging problem at first glance. However, with a systematic approach and the right techniques, we can simplify the expression and find the limit.

Understanding the Problem

The given limit is $\lim _{t \rightarrow 11} \frac{t-11}{\sqrt{t+5}-4}$ . This means that we need to find the value of the expression as $t$ approaches 11. The expression involves a square root, which can be simplified using algebraic manipulations.

Simplifying the Expression

To simplify the expression, we can start by rationalizing the denominator. This involves multiplying the numerator and denominator by the conjugate of the denominator, which is $\sqrt{t+5}+4$ . This will eliminate the square root in the denominator.

\lim _{t \rightarrow 11} \frac{t-11}{\sqrt{t+5}-4} = \lim _{t \rightarrow 11} \frac{(t-11)(\sqrt{t+5}+4)}{(\sqrt{t+5}-4)(\sqrt{t+5}+4)}

Expanding the Denominator

Now, we can expand the denominator using the difference of squares formula: $(a-b)(a+b) = a^2 - b^2$ . In this case, we have $(\sqrt{t+5}-4)(\sqrt{t+5}+4) = (\sqrt{t+5})^2 - 4^2$ .

\lim _{t \rightarrow 11} \frac{(t-11)(\sqrt{t+5}+4)}{(\sqrt{t+5})^2 - 4^2}

Simplifying the Expression Further

Now, we can simplify the expression further by expanding the numerator and denominator.

\lim _{t \rightarrow 11} \frac{(t-11)(\sqrt{t+5}+4)}{t+5-16} = \lim _{t \rightarrow 11} \frac{(t-11)(\sqrt{t+5}+4)}{t-11}

Canceling Out the Common Factor

Now, we can cancel out the common factor of $(t-11)$ in the numerator and denominator.

\lim _{t \rightarrow 11} \frac{(t-11)(\sqrt{t+5}+4)}{t-11} = \lim _{t \rightarrow 11} (\sqrt{t+5}+4)

Evaluating the Limit

Now, we can evaluate the limit by substituting $t=11$ into the expression.

\lim _{t \rightarrow 11} (\sqrt{t+5}+4) = \sqrt{11+5}+4 = \sqrt{16}+4 = 4+4 = 8

Conclusion

In this article, we evaluated the limit of the expression $\lim _{t \rightarrow 11} \frac{t-11}{\sqrt{t+5}-4}$ using a systematic approach and algebraic manipulations. We simplified the expression by rationalizing the denominator, expanding the denominator, and canceling out the common factor. Finally, we evaluated the limit by substituting $t=11$ into the expression. The result is $\boxed{8}$ .

Common Mistakes to Avoid

When evaluating limits, it's essential to avoid common mistakes such as:

Not rationalizing the denominator
Not expanding the denominator
Not canceling out the common factor
Not substituting the correct value of $t$ into the expression

By avoiding these mistakes, you can ensure that your calculations are accurate and your results are reliable.

Real-World Applications

Limits have numerous real-world applications in fields such as physics, engineering, and economics. For example, limits are used to model population growth, electrical circuits, and financial markets. In addition, limits are used to derive many mathematical formulas and equations, such as the derivative of a function.

Final Thoughts

Introduction

In our previous article, we evaluated the limit of the expression $\lim _{t \rightarrow 11} \frac{t-11}{\sqrt{t+5}-4}$ using a systematic approach and algebraic manipulations. In this article, we will answer some frequently asked questions about evaluating limits.

Q: What is a limit?

A: A limit is a value that a function approaches as the input (or independent variable) gets arbitrarily close to a certain point. In other words, it's the value that the function gets arbitrarily close to as the input gets arbitrarily close to a certain point.

Q: Why are limits important?

A: Limits are important because they help us understand how functions behave as the input gets arbitrarily close to a certain point. This is crucial in many areas of mathematics, science, and engineering, such as calculus, physics, and engineering.

Q: How do I evaluate a limit?

A: To evaluate a limit, you need to follow these steps:

Rationalize the denominator: If the denominator contains a square root, multiply the numerator and denominator by the conjugate of the denominator.
Expand the denominator: Use the difference of squares formula to expand the denominator.
Cancel out the common factor: If there is a common factor in the numerator and denominator, cancel it out.
Substitute the correct value of the input: Substitute the value of the input into the expression.

Q: What are some common mistakes to avoid when evaluating limits?

A: Some common mistakes to avoid when evaluating limits include:

Not rationalizing the denominator
Not expanding the denominator
Not canceling out the common factor
Not substituting the correct value of the input
Not checking for undefined values

Q: How do I know if a limit is undefined?

A: A limit is undefined if the denominator is equal to zero, or if the function approaches infinity or negative infinity as the input gets arbitrarily close to a certain point.

Q: Can I use a calculator to evaluate a limit?

A: Yes, you can use a calculator to evaluate a limit. However, it's essential to check the calculator's settings and ensure that it is set to evaluate limits correctly.

Q: Are limits only used in calculus?

A: No, limits are used in many areas of mathematics, science, and engineering, including calculus, physics, engineering, and economics.

Q: Can I use limits to solve real-world problems?

A: Yes, limits can be used to solve real-world problems in many areas, such as:

Modeling population growth
Analyzing electrical circuits
Understanding financial markets
Deriving mathematical formulas and equations

Q: How do I practice evaluating limits?

A: To practice evaluating limits, try the following:

Start with simple limits and gradually move to more complex ones.
Use online resources, such as limit calculators and worksheets.
Practice evaluating limits with different types of functions, such as polynomial, rational, and trigonometric functions.
Join a study group or find a study partner to practice evaluating limits together.

Conclusion

Evaluating limits is a crucial skill for any mathematician, and it requires a systematic approach and algebraic manipulations. By following the steps outlined in this article and avoiding common mistakes, you can become proficient in evaluating limits and solving mathematical problems. Remember to practice regularly and apply the concepts to real-world problems. With patience and persistence, you can master the art of evaluating limits.