How to find limits of rational functions?
Understand the Problem
The question is asking for the methodology or steps required to find limits of rational functions in calculus, particularly as the variable approaches a certain value.
Answer
The limit is $2$.
Answer for screen readers
The limit of the function $f(x) = \frac{x^2 - 1}{x - 1}$ as $x$ approaches 1 is 2, or $\lim_{x \to 1} f(x) = 2$.
Steps to Solve
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Identify the Function and Value to Approach
First, write down the rational function you need to find the limit for, and the value that the variable is approaching. For example, consider the function $f(x) = \frac{x^2 - 1}{x - 1}$ as $x$ approaches 1. -
Substitute the Value
Next, substitute the value you are approaching into the function. If we substitute $x = 1$ into our example:
$$ f(1) = \frac{1^2 - 1}{1 - 1} = \frac{0}{0} $$
This results in an indeterminate form ($\frac{0}{0}$). -
Factor the Function
Now, factor the function to simplify it. In our example:
$$ f(x) = \frac{(x - 1)(x + 1)}{x - 1} $$
For $x \neq 1$, this can be simplified to:
$$ f(x) = x + 1 $$ -
Simplify and Evaluate the Limit
Now, simplify the function by canceling out the common factors. After cancelling, we can evaluate the limit by substituting the value again:
$$ \lim_{x \to 1} f(x) = 1 + 1 = 2 $$ -
State the Limit
Finally, clearly state the limit you have found. For our example, we conclude that:
$$ \lim_{x \to 1} f(x) = 2 $$
The limit of the function $f(x) = \frac{x^2 - 1}{x - 1}$ as $x$ approaches 1 is 2, or $\lim_{x \to 1} f(x) = 2$.
More Information
The limit of rational functions helps to understand the behavior of the function as it approaches a certain value, especially when direct substitution gives an indeterminate form. Factoring and simplifying is a crucial technique in calculus to resolve limits effectively.
Tips
- Not Checking for Indeterminate Forms: Always check if you encounter a $\frac{0}{0}$ form before proceeding with factoring.
- Forgetting to Simplify: Make sure to simplify the equation before substituting again after factoring, as this allows for proper evaluation of the limit.
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