Final answer:
To solve the limit using L'Hôpital's Rule, we find the derivatives of the numerator and denominator. As x approaches negative infinity, the original expression is an indeterminate form of ∞/∞, allowing us to apply the rule. After taking derivatives, the limit as x approaches negative infinity is 3/4.
Step-by-step explanation:
Using L'Hôpital's Rule to Solve a Limit
To solve the limit lim x→-∞ (3x+7) / √(16x2−5), we must apply L'Hôpital's Rule which states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form such as 0/0 or ∞/∞, then the limit can be found by taking the derivative of the numerator and the denominator separately and then finding the limit of their quotient.
Step 1: Identify Indeterminate Form
As x approaches negative infinity, the terms 3x and √(16x2) both approach negative infinity, which indicates an indeterminate form of ∞/∞. Therefore, we can apply L'Hôpital's Rule.
Step 2: Differentiate Numerator and Denominator
The derivative of 3x is 3, and the derivative of √(16x2−5) using the chain rule is √(16x2−5)−1 × d/dx (16x2−5) = (32x)/(2√(16x2−5)).
Step 3: Apply L'Hôpital's Rule
We take the limit of the derivatives' quotient: lim x→-∞ 3 / ((32x)/(2√(16x2−5))). Simplifying, we get the limit as lim x→-∞ (3√(16x2−5))/(32x), which simplifies further to lim x→-∞ (√(16x2)/x). Since √(16x2)/x equals 4 when x is negative, the limit is 3/4.