Final answer:
Direct substitution is a method used to evaluate limits in calculus. It involves plugging the given value of the variable directly into the function and simplifying to find the limit. Sometimes, this method leads to an indeterminate form.
Step-by-step explanation:
Direct substitution is a method used to evaluate limits in calculus. It involves plugging the given value of the variable directly into the function and simplifying to find the limit. However, sometimes this method leads to an indeterminate form, where the result is not immediately clear.
One common indeterminate form is 0/0, where both the numerator and denominator of the function approach 0 as the variable approaches the given value. Another indeterminate form is ∞/∞, where both the numerator and denominator approach infinity. Another indeterminate form is 1^∞, where the function has the form of a power of 1 raised to infinity.
For example, if you have the function f(x) = (x^2 - 1) / (x - 1) and you want to evaluate the limit as x approaches 1, direct substitution gives you f(1) = 0/0, which is an indeterminate form. To evaluate this limit, you would need to use other techniques, such as factoring or rationalizing the numerator and denominator.