Problem 35
Question
In Problems 29-48, find the limits. $$ \lim _{x \rightarrow-1} \sqrt{x^{2}+2 x+2} $$
Step-by-Step Solution
Verified Answer
The limit is 1.
1Step 1: Understand the limit expression
We are asked to find the limit of the function \( \sqrt{x^2 + 2x + 2} \) as \( x \) approaches \(-1\). This means we want to determine the value the function approaches when \( x \) gets close to \(-1\).
2Step 2: Evaluate the function at the limit point
Substitute \( x = -1 \) into the expression inside the square root: \((-1)^2 + 2(-1) + 2\). This simplifies to \(1 - 2 + 2 = 1\).
3Step 3: Apply the square root to the result
Take the square root of 1: \( \sqrt{1} = 1 \).
4Step 4: State the limit
Since the function \( \sqrt{x^2 + 2x + 2} \) is continuous and defined around \( x = -1 \), the limit can be directly determined from the substitution. Thus, the limit of the function as \( x \) approaches \(-1\) is 1.
Key Concepts
ContinuityLimit EvaluationFunction Behavior Near a Point
Continuity
Continuity in calculus refers to a function that is smooth and unbroken at a point and around it. This means you can draw its graph without lifting a pencil. A continuous function does not have any jumps, breaks, or holes. When we talk about the continuity of a function, we're interested in whether the function's limit at a point equals the actual value of the function at that point.
For the function \( \sqrt{x^2 + 2x + 2} \), it is continuous for all real values of \( x \) because it involves basic operations (addition, subtraction, and square roots) on continuous functions (polynomials). A key point to understand is that if a function is continuous at a point, you can find the limit simply by substitution.
This makes evaluating limits straightforward for continuous functions, such as in our specific problem where we evaluated the limit as \( x \) approaches -1 by direct substitution.
For the function \( \sqrt{x^2 + 2x + 2} \), it is continuous for all real values of \( x \) because it involves basic operations (addition, subtraction, and square roots) on continuous functions (polynomials). A key point to understand is that if a function is continuous at a point, you can find the limit simply by substitution.
This makes evaluating limits straightforward for continuous functions, such as in our specific problem where we evaluated the limit as \( x \) approaches -1 by direct substitution.
Limit Evaluation
Limit evaluation involves finding out what value a function approaches as the input gets close to a specified point. To find a limit, imagine sliding the values of \( x \) closer and closer to the point of interest and see what happens to the value of the function.
For the given expression \( \sqrt{x^2 + 2x + 2} \), as \( x \) approaches -1, substitute -1 in the function: \((-1)^2 + 2(-1) + 2\). This evaluates to 1, giving us \( \sqrt{1} = 1\).
Sometimes, finding limits require more complex techniques like factoring, rationalizing, or using L'Hôpital's Rule. However, for many continuous functions, direct substitution suffices. Verifying continuity is often the first step to applying direct substitution effectively.
For the given expression \( \sqrt{x^2 + 2x + 2} \), as \( x \) approaches -1, substitute -1 in the function: \((-1)^2 + 2(-1) + 2\). This evaluates to 1, giving us \( \sqrt{1} = 1\).
Sometimes, finding limits require more complex techniques like factoring, rationalizing, or using L'Hôpital's Rule. However, for many continuous functions, direct substitution suffices. Verifying continuity is often the first step to applying direct substitution effectively.
Function Behavior Near a Point
Understanding the behavior of a function near a point enhances our insight into its graph and values. It involves observing how the function behaves and what it approaches as \( x \) nears a particular value, like -1 in our example.
For \( \sqrt{x^2 + 2x + 2} \), it's helpful to consider slightly varying \( x \) around -1: values like -1.1, -0.9, etc. These values, when plugged into the function, will show how the function output changes, inching closer to the limit value of 1.
This examination reassures us that there are no unexpected jumps or changes. Function behavior analysis might reveal critical points or asymptotic behaviors, which are essential in graphing and in understanding the full scope of the function's properties.
For \( \sqrt{x^2 + 2x + 2} \), it's helpful to consider slightly varying \( x \) around -1: values like -1.1, -0.9, etc. These values, when plugged into the function, will show how the function output changes, inching closer to the limit value of 1.
This examination reassures us that there are no unexpected jumps or changes. Function behavior analysis might reveal critical points or asymptotic behaviors, which are essential in graphing and in understanding the full scope of the function's properties.
Other exercises in this chapter
Problem 34
In Problems 29-48, find the limits. $$ \lim _{x \rightarrow-2} \sqrt{6+x} $$
View solution Problem 34
Use a table and a graph to find out what happens to $$ f(x)=\exp \left(\frac{1}{x}\right) $$ as \(x \rightarrow 0\)
View solution Problem 35
Use a graphing calculator to investigate $$ \lim _{x \rightarrow 0} \frac{\sin x}{x} $$
View solution Problem 36
In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 1} \sqrt{x^{3}+4 x-1} $$
View solution