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Chapter 4: Problem 6

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. \(\lim _{t \rightarrow 0} \frac{e^{3 t}-1}{t}\)

Short Answer

Expert verified
The limit is 3.

Step by step solution

01

Identify the Indeterminate Form

First, plug in the value of \( t = 0 \) in the expression. This gives \( \frac{e^{3 \cdot 0} - 1}{0} = \frac{1 - 1}{0} = \frac{0}{0} \). This is an indeterminate form, which means we can consider using l'Hospital's Rule.
02

Apply l'Hospital's Rule

Since the expression is in the form \( \frac{0}{0} \), l'Hospital's Rule can be used, which states that \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \) if this limit exists. Here, set \( f(t) = e^{3t} - 1 \) and \( g(t) = t \).
03

Differentiate Numerator and Denominator

Calculate the derivative of the numerator: \( f'(t) = \frac{d}{dt}[e^{3t} - 1] = 3e^{3t} \). Calculate the derivative of the denominator: \( g'(t) = \frac{d}{dt}[t] = 1 \).
04

Apply the Derivatives in the Limit

Now apply l'Hospital's Rule: \( \lim_{t \to 0} \frac{3e^{3t}}{1} = 3e^{3 \cdot 0} = 3 \cdot 1 = 3 \).
05

Conclusion

Since the limit using l'Hospital's Rule results in a finite number, \( 3 \), we have successfully found the limit.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

L'Hospital's Rule
L'Hospital's Rule is a powerful tool in calculus, designed to help find limits that present indeterminate forms. Often, when calculating limits, we run into expressions like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These are called indeterminate forms because they don't provide a clear answer about the value of a limit just by initial substitution. L'Hospital's Rule provides a method for resolving these by taking derivatives.

Here's how it works:
  • Identify the indeterminate form. Common forms include \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
  • Ensure the limit itself is valid for L'Hospital's Rule. This means both the numerator and denominator must be differentiable near the point of interest.
  • Apply the rule: Replace the limit of the original fraction with the limit of the fraction of their derivatives. In other words, if \( \lim_{x \to a} \frac{f(x)}{g(x)} \) results in an indeterminate form, try \( \lim_{x \to a} \frac{f'(x)}{g'(x)} \).
  • Finally, calculate the new limit. If it is still indeterminate, L'Hospital's Rule may be applied repeatedly if necessary.
Remember, this rule simplifies the process of finding limits, but it's only applicable under specific conditions.
Limits
Limits are foundational in calculus, capturing the behavior of a function as it approaches a specific point. The notation \( \lim_{x \to a} f(x) \) helps us understand what value \( f(x) \) approaches as \( x \) gets closer to \( a \). Calculating limits involves predicting the value a function seems to be heading towards without necessarily reaching it.

Understanding limits enhances comprehension of functions' overall behavior. They aid in:
  • Determining continuity by checking if \( \lim_{x \to a} f(x) = f(a) \).
  • Calculating derivatives, which are expressed as a limit themselves.
  • Identifying asymptotic behavior, understanding functions near undefined points or infinity.
Limits not only help in solving specific problems but also underlie many broader concepts in calculus, making a strong grasp of them essential for deeper mathematical studies.
Indeterminate Forms
Indeterminate Forms cause difficulties when calculating limits, often giving expressions like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These forms don't have a simple numerical value, leading to ambiguities. When faced with such forms, direct substitution may not yield meaningful results.

To solve limits involving indeterminate forms, one can:
  • Use algebraic manipulation to simplify the expression.
  • Apply L'Hospital's Rule where appropriate to differentiate the numerator and denominator until a clear limit value emerges.
  • Factor or rationalize expressions to remove the indeterminate form.
Recognizing indeterminate forms is crucial for effectively resolving them, turning unclear limits into clear ones.

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