Technique for Evaluating Limits | Basic Limit Substitution in Calculus

Basic limit substitution

Basic limit substitution is a technique used in calculus to evaluate limits

Basic limit substitution is a technique used in calculus to evaluate limits. It involves substituting a specific value or variable in the expression for the limiting variable, in order to simplify the expression and evaluate the limit.

The general steps for performing limit substitution are as follows:

1. Identify the expression for which you want to evaluate the limit.
2. Check if the expression is in an indeterminate form, such as 0/0 or ∞/∞. If it is not in an indeterminate form, you can simply evaluate the expression.
3. If the expression is in an indeterminate form, choose a substitution that could simplify the expression. Generally, this involves looking for rational functions or expressions containing radicals or trigonometric functions.
4. Substitute the limiting variable with the chosen substitution.
5. Simplify the expression by algebraic manipulation or by applying known limits for the substitution.
6. Finally, substitute the limiting variable back into the expression to obtain the evaluated limit.

Let’s illustrate this with an example:

Example: Evaluate the limit as x approaches 0 of (sin(3x))/(4x).

1. Expression: (sin(3x))/(4x)
2. Check for indeterminate form: This expression is in the indeterminate form 0/0, so we proceed with limit substitution.
3. Substitution: Let’s substitute 3x = u, which implies x = u/3.
4. Simplify: Rewriting the expression, we have (sin(u))/(4(u/3)) = (sin(u))/(4u/3) = (3sin(u))/(4u).
5. Simplifying further: sin(u)/(u) is a known limit, which evaluates to 1. So, we have (3sin(u))/(4u) = (3/4)*(1) = 3/4.
6. Substituting back: We substituted 3x for u, so substituting back gives us 3/4 as the final answer.

Therefore, the limit of (sin(3x))/(4x) as x approaches 0 is 3/4.

More Answers: