Estimate the limit numerically or state that the limit does...

Estimate the limit numerically or state that the limit does not exist. If infinite, state whether the one-sided limits are $\infty$ or $-\infty .$
\begin{equation}\lim _{t \rightarrow e} \frac{t-e}{\ln t-1}\end{equation}

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Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which measures the rate at which the function's value changes with respect to changes in its input. It plays a critical role in applying L'Hôpital's Rule, as the rule requires the calculation of derivatives to resolve indeterminate limits.

L'Hôpital's Rule

L'Hôpital's Rule is a method used in calculus to evaluate the limits that result in indeterminate forms like 0/0 or ?/?. The rule states that under appropriate conditions, the limit of a ratio of two functions can be found by taking the limit of the ratio of their derivatives, often simplifying the evaluation process.

Limits

Limits are fundamental concepts in calculus that describe the value that a function approaches as its input approaches a certain point. They serve as the foundation for defining derivatives and integrals and are pivotal in understanding continuous behavior in mathematical functions.

Indeterminate Forms

Indeterminate forms occur when the direct substitution in a limit computation yields an ambiguous expression, such as 0/0 or ?/?. Recognizing indeterminate forms is crucial because it signals that additional techniques, like algebraic manipulation or applying L'Hôpital's Rule, are necessary to evaluate the limit.

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