Find the limit. Use l'Hospital's Rule where appropriate. If...

Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.

$ \displaystyle \lim_{x\to 1/2} \frac{6x^2 + 5x - 4}{4x^2 + 16x - 9} $

Key Concepts

Limits

Limits are a fundamental concept in calculus that describe the value a function approaches as the input approaches a particular point. This concept is used to analyze the behavior of functions at points where they may not be explicitly defined or where they show interesting behavior such as asymptotes or discontinuities.

Indeterminate Forms

Indeterminate forms occur when the direct substitution in a limit results in an undefined expression like 0/0, ?/?, or other forms that do not immediately reveal the limit. Recognizing these forms is essential because it indicates that further analysis, such as algebraic manipulation or advanced methods, is needed to evaluate the limit.

l'Hôpital's Rule

l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, particularly 0/0 or ?/?. The rule states that if the limits of the numerator and denominator both approach 0 (or both approach infinity), the limit of the quotient can be found by taking the derivative of the numerator and the derivative of the denominator, provided these derivatives exist and the new limit exists or is ? or -?.

Derivatives

Derivatives represent the rate of change of a function with respect to its variable and are a core concept in calculus. They are used not only to determine the slopes of tangents to curves but also in methods like l'Hôpital's Rule, where the derivatives of the numerator and denominator are used to resolve indeterminate forms in limits.

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