Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
85. lim(x→1) (x² + 3x - 4)/(x - 1)

Verified step by step guidance

1. First, check if the limit results in an indeterminate form by substituting \(x = 1\) into the expression \(\frac{x^{2} + 3x - 4}{x - 1}\). If both numerator and denominator approach 0, then l'Hôpital's Rule can be applied.
2. Since direct substitution gives \(\frac{1^{2} + 3(1) - 4}{1 - 1} = \frac{0}{0}\), which is an indeterminate form, proceed with l'Hôpital's Rule.
3. Apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \(x\). The derivative of the numerator \(x^{2} + 3x - 4\) is \(2x + 3\), and the derivative of the denominator \(x - 1\) is \(1\).
4. Rewrite the limit using these derivatives: \(\lim_{x \to 1} \frac{2x + 3}{1}\).
5. Finally, evaluate the new limit by substituting \(x = 1\) into the simplified expression \(\frac{2(1) + 3}{1}\) to find the limit.

Key Concepts

Limits and Indeterminate Forms

Limits describe the behavior of a function as the input approaches a certain value. When direct substitution results in an indeterminate form like 0/0, special techniques such as l’Hôpital’s Rule are needed to evaluate the limit.

l’Hôpital’s Rule

l’Hôpital’s Rule states that if a limit results in an indeterminate form 0/0 or ∞/∞, the limit of the ratio of the functions equals the limit of the ratio of their derivatives, provided this latter limit exists.

Derivative of Functions

The derivative measures the instantaneous rate of change of a function. To apply l’Hôpital’s Rule, you must differentiate the numerator and denominator separately and then evaluate the new limit.