Mastering the Four Methods for Evaluating Limits in Mathematics: Direct Substitution, Factoring and Canceling, Finding Common Denominator, and L’Hospital’s Rule

4 Ways to Evaluate a Limit

To evaluate a limit in mathematics, there are generally four common methods that can be used

To evaluate a limit in mathematics, there are generally four common methods that can be used. These methods are direct substitution, factoring and canceling, finding common denominator, and using L’Hospital’s rule. Let’s explore each method in detail.

1. Direct Substitution:
This method is the simplest and most basic way to evaluate a limit. It involves substituting the value in question directly into the function and simplifying. However, direct substitution can only be used when the resulting expression is defined.

Example: Evaluate the limit as x approaches 2 for the function f(x) = x^2 + 3x – 10.
We substitute x = 2 into the function:
f(2) = (2)^2 + 3(2) – 10
= 4 + 6 – 10
= 0
Therefore, the limit is 0.

2. Factoring and Canceling:
This method is useful when you have a rational function with common factors in the numerator and denominator. By factoring out common terms and canceling them, it is often possible to simplify the expression and evaluate the limit.

Example: Evaluate the limit as x approaches 4 for the function g(x) = (x^2 – 16)/(x – 4).
We notice that the numerator is a difference of squares, so it can be factored:
g(x) = [(x – 4)(x + 4)]/(x – 4)
By canceling the common factor (x – 4), we have:
g(x) = x + 4
Now we can substitute x = 4 into the simplified expression:
g(4) = 4 + 4
= 8
Thus, the limit is 8.

3. Finding Common Denominator:
This method is applicable when you encounter a fraction with different denominators. By finding the least common denominator (LCD), you can combine the fractions into a single expression, which can often simplify and allow for evaluating the limit.

Example: Evaluate the limit as x approaches 0 for the function h(x) = (1/x) + (3/x^2).
To find the common denominator, we multiply the first fraction by x/x:
h(x) = (1/x) * (x/x) + (3/x^2)
= (x + 3/x^2)
Now, we can substitute x = 0 into the simplified expression (note that the denominator should not be equal to 0):
h(0) = 0 + 3/0^2
= undefined
Hence, the limit does not exist.

4. L’Hospital’s Rule:
L’Hospital’s rule is used when evaluating limits involving indeterminate forms, such as 0/0 or ∞/∞. It allows for computing limits by taking the derivative of the numerator and denominator and then evaluating the limit again.

Example: Evaluate the limit as x approaches ∞ for the function f(x) = x/e^x.
The limit is an indeterminate form (∞/∞), so we can apply L’Hospital’s rule:
Take the derivative of the numerator and denominator:
f'(x) = (1) / (e^x)
Now, we can evaluate the limit of the derivative:
lim(x→∞) [(1) / (e^x)]
As x approaches ∞, e^x also approaches ∞,
lim(x→∞) [(1) / (∞)] = 0
Therefore, the original limit is 0.

These four methods provide different approaches to evaluate limits in mathematics. It is important to select the most suitable method based on the form of the function and the desired outcome.

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