A Guide to Calculating Limits: Techniques and Examples

Limit equation

A limit can be defined as the value that a function approaches as the input approaches a certain value

A limit can be defined as the value that a function approaches as the input approaches a certain value. It helps us understand the behavior of a function near a particular point.

The notation for a limit is as follows:

lim (x -> a) f(x)

This means “the limit of f(x) as x approaches a.”

To calculate a limit, there are different techniques you can use depending on the type of function. Let’s go over a few common cases:

1. Polynomial Functions:
For a polynomial function, we can find the limit by simply substituting the value of the variable into the function. For example, if we have the function f(x) = 2x^2 + 3x – 1, and we want to find the limit as x approaches 2, we substitute 2 into the equation:

lim (x->2) (2x^2 + 3x – 1)
= 2(2)^2 + 3(2) – 1
= 2(4) + 6 – 1
= 8 + 6 – 1
= 13

So, the limit of f(x) as x approaches 2 is 13.

2. Rational Functions:
For rational functions (a quotient of two polynomials), we need to be careful if the denominator becomes zero at the point we are approaching. In that case, the limit may not exist or may be infinite.
For example, let’s say we have the function g(x) = (x^2 – 3x + 2)/(x – 2), and we want to find the limit as x approaches 2.

lim (x->2) [(x^2 – 3x + 2)/(x – 2)]
= [(2^2 – 3(2) + 2)/(2 – 2)]
= [(4 – 6 + 2)/(0)]
= 0/0

In this case, we get an indeterminate form of 0/0. To resolve this, we can use algebraic techniques like factoring, simplifying, or multiplying and dividing by the conjugate to manipulate the equation into a form where we can substitute the value of x. After simplification, we can find that the limit is 3.

3. Trigonometric Functions:
For trigonometric functions, we use the fundamental limits where x approaches zero, such as sin(x)/x = 1, cos(x) = 1, and tan(x)/x = 1, to evaluate limits at various points.

For example, if we have the function h(x) = (sin(x))/x, and we want to find the limit as x approaches 0:

lim (x->0) (sin(x))/x
= 1

Therefore, the limit is 1 as x approaches 0.

These are just a few examples of how to calculate limits. Pay attention to the properties and special cases of different functions to determine the appropriate method for finding the limit. Remember, practice and familiarity with various functions will help you become more comfortable and proficient in evaluating limits.

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