Problem 20
Question
Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 1}(-2 x+5)$$
Step-by-Step Solution
Verified Answer
Answer: The limit of the function as x approaches 1 is 3.
1Step 1: Identify the function
The given function is:
$$f(x) = -2x + 5$$
2Step 2: Find the limit
We need to find the limit as x approaches 1:
$$\lim_{x \rightarrow 1}(-2x + 5)$$
Since the function is linear, the limit exists, and it's equal to the value of the function at x=1.
3Step 3: Evaluate the function at x=1
Now, substitute x = 1 into the function:
$$f(1) = -2(1) + 5 = -2 + 5 = 3$$
4Step 4: State the answer
So, the limit of the function as x approaches 1 is 3:
$$\lim_{x \rightarrow 1}(-2x + 5) = 3$$
Key Concepts
Finding LimitsLimits of Linear FunctionsLimit EvaluationCalculus Problem Solving
Finding Limits
The process of finding limits in calculus involves determining what value a function approaches as the input (usually denoted by the variable x) gets arbitrarily close to a certain number. Finding a limit can reveal the behavior of a function around a specific point or at infinity. It's a fundamental concept because it helps us understand continuity and predict function values that aren't easily computed directly.
For instance, in the exercise presented, we are asked to find the limit of the function \(f(x) = -2x + 5\) as \(x\) approaches 1. To find this limit, we follow steps that help us deduce this approaching value, starting from identifying the given function, proceeding to apply limit laws if necessary, and finally evaluating the function at a specific point when applicable.
For instance, in the exercise presented, we are asked to find the limit of the function \(f(x) = -2x + 5\) as \(x\) approaches 1. To find this limit, we follow steps that help us deduce this approaching value, starting from identifying the given function, proceeding to apply limit laws if necessary, and finally evaluating the function at a specific point when applicable.
Limits of Linear Functions
Linear functions are the simplest types of functions when it comes to limit evaluation. They have the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants, representing the slope and y-intercept, respectively. Limits of linear functions are straightforward because as \(x\) approaches any real number, the function directly approaches the value \(mx + b\) at that point.
Therefore, to find the limit of a linear function as \(x\) approaches any number, we simply substitute that number into the function. This direct substitution is possible because linear functions are continuous everywhere—they do not have breaks or jumps.
Therefore, to find the limit of a linear function as \(x\) approaches any number, we simply substitute that number into the function. This direct substitution is possible because linear functions are continuous everywhere—they do not have breaks or jumps.
Limit Evaluation
Limit evaluation is a critical skill in calculus and it comes with a set of strategies, including direct substitution, factoring, rationalizing, and applying special limit laws. In the context of linear functions, limit evaluation is typically as simple as finding the function's value at a specific point since they are continuous and do not pose complex behaviors near the point of interest.
When evaluating the limit of a function that isn't linear, direct substitution might not always work, as some functions have undefined points or indeterminate forms like \(0/0\). In such cases, we might need to simplify the function or use other techniques such as L'Hôpital's rule, which requires finding the derivative of the function's numerator and denominator.
When evaluating the limit of a function that isn't linear, direct substitution might not always work, as some functions have undefined points or indeterminate forms like \(0/0\). In such cases, we might need to simplify the function or use other techniques such as L'Hôpital's rule, which requires finding the derivative of the function's numerator and denominator.
Calculus Problem Solving
Calculus problem solving involves a mixture of analytical skills and a deep understanding of calculus principles. When faced with a calculus problem, it's important to break it down into smaller, manageable steps. Start by understanding what the problem is asking, then identify the type of function you are dealing with, and choose the right strategy for limit evaluation.
Remember that not every function's limit can be found through direct substitution. It's essential to be familiar with various problem-solving techniques to handle different types of functions and scenarios. Finally, always verify your answer to ensure it makes sense within the context of the problem. Good practice and a methodical approach can significantly enhance your proficiency in calculus problem solving.
Remember that not every function's limit can be found through direct substitution. It's essential to be familiar with various problem-solving techniques to handle different types of functions and scenarios. Finally, always verify your answer to ensure it makes sense within the context of the problem. Good practice and a methodical approach can significantly enhance your proficiency in calculus problem solving.
Other exercises in this chapter
Problem 19
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