Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim t→5 (1/t^2 − 4t − 5 −1/ 6(t − 5))

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Identify the limit expression: \( \lim_{{t \to 5}} \left( \frac{1}{t^2 - 4t - 5} - \frac{1}{6(t - 5)} \right) \).

Factor the quadratic expression in the denominator: \( t^2 - 4t - 5 = (t - 5)(t + 1) \).

Rewrite the limit expression using the factored form: \( \lim_{{t \to 5}} \left( \frac{1}{(t - 5)(t + 1)} - \frac{1}{6(t - 5)} \right) \).

Combine the fractions over a common denominator: \( \frac{1}{(t - 5)(t + 1)} - \frac{1}{6(t - 5)} = \frac{6 - (t + 1)}{6(t - 5)(t + 1)} \).

Simplify the numerator: \( 6 - (t + 1) = 5 - t \), and then evaluate the limit as \( t \to 5 \).

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주요 개념

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Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this question, we are tasked with finding the limit of a function as t approaches 5, which requires evaluating the function's behavior close to that point.

Indeterminate Forms

Indeterminate forms occur in calculus when direct substitution into a limit results in expressions like 0/0 or ∞/∞. These forms require further analysis, often using algebraic manipulation or L'Hôpital's Rule, to resolve the limit. In the given question, substituting t = 5 directly into the expression leads to an indeterminate form, necessitating additional steps to find the limit.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms by differentiating the numerator and denominator. If a limit results in 0/0 or ∞/∞, applying this rule can simplify the expression and help find the limit. In this case, if the limit leads to an indeterminate form, L'Hôpital's Rule may be a suitable approach to determine the limit as t approaches 5.

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.lim t→5 (1/t^2 − 4t − 5 −1/ 6(t − 5))