Solving Limits of 0/0 or Infinity/Infinity | L’Hôpital’s Rule, Algebraic Manipulation, and Rewriting Techniques

When a limit equals 0/0 or infinity/infinity, what should I do?

When faced with a limit that evaluates to 0/0 or infinity/infinity, you can use various mathematical techniques to solve these indeterminate forms

When faced with a limit that evaluates to 0/0 or infinity/infinity, you can use various mathematical techniques to solve these indeterminate forms. The most commonly used techniques are L’Hôpital’s Rule, algebraic manipulation, and rewriting the function.

1. L’Hôpital’s Rule: This rule allows you to evaluate limits of the form 0/0 or infinity/infinity by taking derivatives of the numerator and denominator until you obtain an expression whose limit can be determined easily. To apply L’Hôpital’s Rule, follow these steps:
a. Take the derivatives of both the numerator and denominator.
b. Evaluate the limit of the derivative of the numerator divided by the derivative of the denominator.
c. If the limit still evaluates to 0/0 or infinity/infinity, repeat the process until you obtain a determinable limit.

2. Algebraic Manipulation: Sometimes, you can manipulate the expression algebraically to simplify it before evaluating the limit. Common techniques include factoring, rationalizing, or using trigonometric identities to simplify the expression. By simplifying the expression, you may be able to cancel out common factors, which can help determine the limit.

3. Rewriting the Function: Another approach is to rewrite the function using equivalent expressions that are easier to evaluate. For example, you can multiply or divide the expression by a suitable conjugate or use the properties of logarithms or exponentials to simplify the expression. By rewriting the function, you may be able to transform the limit into a form that can be easily determined.

Remember that not all limits of the form 0/0 or infinity/infinity can be solved using these techniques. In some cases, you may need to explore other methods such as series expansions or using special trigonometric identities. Additionally, it’s important to carefully analyze the problem and determine if the limit actually exists or if it is undefined.

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