Evaluate The Limit:$\lim _{t \rightarrow 3} \frac{\sqrt{19-t}-4}{t-3}$

Introduction

Limits are a fundamental concept in calculus, and evaluating them is a crucial skill for any math student or professional. In this article, we will focus on evaluating the limit of a specific function, $\lim _{t \rightarrow 3} \frac{\sqrt{19-t}-4}{t-3}$. We will break down the problem into manageable steps, using algebraic manipulations and mathematical techniques to simplify the expression and find the limit.

Understanding the Problem

The given limit is $\lim _{t \rightarrow 3} \frac{\sqrt{19-t}-4}{t-3}$. This is a limit of a rational function, where the numerator is a square root expression and the denominator is a linear expression. The limit is taken as $t$ approaches $3$.

Step 1: Simplify the Expression

To evaluate the limit, we first need to simplify the expression. We can start by rationalizing the numerator, which involves multiplying the numerator and denominator by the conjugate of the numerator.

\lim _{t \rightarrow 3} \frac{\sqrt{19-t}-4}{t-3} = \lim _{t \rightarrow 3} \frac{\sqrt{19-t}-4}{t-3} \cdot \frac{\sqrt{19-t}+4}{\sqrt{19-t}+4}

This simplifies to:

\lim _{t \rightarrow 3} \frac{(19-t)-16}{(t-3)(\sqrt{19-t}+4)}

Step 2: Simplify the Expression Further

We can simplify the expression further by expanding the numerator and denominator.

\lim _{t \rightarrow 3} \frac{3-t}{(t-3)(\sqrt{19-t}+4)} = \lim _{t \rightarrow 3} \frac{3-t}{(t-3)(\sqrt{19-t}+4)}

Step 3: Evaluate the Limit

Now that we have simplified the expression, we can evaluate the limit. As $t$ approaches $3$, the denominator approaches $0$, and the numerator approaches $0$. However, the limit is not equal to $0$ because the denominator is approaching $0$ faster than the numerator.

To evaluate the limit, we can use L'Hopital's rule, which states that if a limit is of the form $\frac{0}{0}$, we can take the derivative of the numerator and denominator and evaluate the limit of the resulting expression.

\lim _{t \rightarrow 3} \frac{3-t}{(t-3)(\sqrt{19-t}+4)} = \lim _{t \rightarrow 3} \frac{-1}{(\sqrt{19-t}+4)+\frac{(t-3)}{2\sqrt{19-t}}}

Step 4: Simplify the Expression Further

We can simplify the expression further by combining the terms in the denominator.

\lim _{t \rightarrow 3} \frac{-1}{(\sqrt{19-t}+4)+\frac{(t-3)}{2\sqrt{19-t}}} = \lim _{t \rightarrow 3} \frac{-1}{\frac{2\sqrt{19-t}+4\sqrt{19-t}+2(t-3)}{2\sqrt{19-t}}}

Step 5: Evaluate the Limit

Now that we have simplified the expression, we can evaluate the limit. As $t$ approaches $3$, the denominator approaches $2\sqrt{19-3}+4\sqrt{19-3}$, which is equal to $6\sqrt{16}$.

\lim _{t \rightarrow 3} \frac{-1}{\frac{2\sqrt{19-t}+4\sqrt{19-t}+2(t-3)}{2\sqrt{19-t}}} = \lim _{t \rightarrow 3} \frac{-1}{\frac{6\sqrt{16}}{2\sqrt{19-t}}}

Conclusion

In this article, we evaluated the limit of a specific function, $\lim _{t \rightarrow 3} \frac{\sqrt{19-t}-4}{t-3}$. We broke down the problem into manageable steps, using algebraic manipulations and mathematical techniques to simplify the expression and find the limit. The final answer is $\boxed{-\frac{1}{6}}$.

Final Answer

Q&A: Evaluating Limits

Q: What is a limit in mathematics?

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

Q: Why is evaluating limits important?

A: Evaluating limits is important because it helps us understand the behavior of functions as the input gets arbitrarily close to a certain point. This is crucial in many areas of mathematics, such as calculus, where limits are used to define the derivative and integral of a function.

Q: What are some common types of limits?

A: Some common types of limits include:

• One-sided limits: These are limits that approach a certain point from one side only.
• Two-sided limits: These are limits that approach a certain point from both sides.
• Infinite limits: These are limits that approach infinity as the input gets arbitrarily close to a certain point.
• Undefined limits: These are limits that do not exist or are undefined.

Q: How do I evaluate a limit?

A: To evaluate a limit, you can use the following steps:

1. Simplify the expression: Try to simplify the expression by combining like terms or canceling out common factors.
2. Use algebraic manipulations: Use algebraic manipulations such as factoring, expanding, or canceling out terms to simplify the expression.
3. Use mathematical techniques: Use mathematical techniques such as L'Hopital's rule or the squeeze theorem to evaluate the limit.
4. Check for undefined limits: Check if the limit is undefined or does not exist.

Q: What is L'Hopital's rule?

A: L'Hopital's rule is a mathematical technique used to evaluate limits of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$. It states that if a limit is of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, you can take the derivative of the numerator and denominator and evaluate the limit of the resulting expression.

Q: What is the squeeze theorem?

A: The squeeze theorem is a mathematical technique used to evaluate limits of the form $\lim _{x \rightarrow a} f(x)$, where $f(x)$ is a function that is bounded by two other functions, $g(x)$ and $h(x)$. It states that if $g(x) \leq f(x) \leq h(x)$ for all $x$ in a certain interval, and if $\lim _{x \rightarrow a} g(x) = \lim _{x \rightarrow a} h(x) = L$, then $\lim _{x \rightarrow a} f(x) = L$.

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

A: A limit is undefined if it does not exist or is not defined. This can happen if the function is not continuous at the point of interest, or if the function approaches infinity or negative infinity as the input gets arbitrarily close to the point of interest.

Conclusion

In this article, we answered some common questions about evaluating limits, including what a limit is, why it's important, and how to evaluate a limit. We also discussed some common types of limits, such as one-sided limits, two-sided limits, infinite limits, and undefined limits. Finally, we discussed some mathematical techniques used to evaluate limits, such as L'Hopital's rule and the squeeze theorem.