Is F(t) = (e^t − 1)/t , tan(t), 1/(t + 1) at t0 = 0 continuous at t0?

To determine if the given functions are continuous at \( t_0 = 0 \), we need to check whether the function approaches a finite limit as \( t \) approaches 0 and if the function itself is defined at \( t_0 = 0 \).

### 1. \( F_1(t) = \frac{e^t - 1}{t} \)

This function has a potential issue at \( t = 0 \) because the denominator becomes 0. So, we need to evaluate the limit as \( t \to 0 \).

\[
\lim_{t \to 0} \frac{e^t - 1}{t}
\]

This is a standard limit, and using L'Hopital's Rule (since it's of the form \( \frac{0}{0} \)):

\[
\lim_{t \to 0} \frac{e^t - 1}{t} = \lim_{t \to 0} \frac{e^t}{1} = e^0 = 1
\]

Thus, the limit exists and equals 1. We can now define \( F_1(0) = 1 \) to make the function continuous at \( t_0 = 0 \).

### 2. \( F_2(t) = \tan(t) \)

The function \( \tan(t) \) is defined and continuous everywhere except at points where \( t = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. Since 0 is not one of these points, the function \( \tan(t) \) is continuous at \( t_0 = 0 \), and we have:

\[
\lim_{t \to 0} \tan(t) = \tan(0) = 0
\]

Thus, \( \tan(t) \) is continuous at \( t_0 = 0 \).

### 3. \( F_3(t) = \frac{1}{t + 1} \)

This function is defined and continuous for all real values of \( t \). At \( t = 0 \):

\[
F_3(0) = \frac{1}{0 + 1} = 1
\]

Thus, \( F_3(t) \) is continuous at \( t_0 = 0 \).

### Conclusion:

- \( F_1(t) = \frac{e^t - 1}{t} \) is **continuous** at \( t_0 = 0 \) if we define \( F_1(0) = 1 \).
- \( F_2(t) = \tan(t) \) is **continuous** at \( t_0 = 0 \).
- \( F_3(t) = \frac{1}{t + 1} \) is **continuous** at \( t_0 = 0 \).

Would you like any further details or explanations?

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Here are 5 related questions:

1. How do you apply L'Hopital's Rule to other limits?
2. Can you find the derivative of \( F_1(t) = \frac{e^t - 1}{t} \)?
3. What is the behavior of \( \tan(t) \) at its discontinuities?
4. How do you verify continuity for more complicated functions at a point?
5. What are the conditions for a function to be continuous at a point?

**Tip**: To check continuity, always verify that the function is defined at the point and that the limit as \( t \to t_0 \) equals the function's value at \( t_0 \).

Is F(t) = (e^t − 1)/t, tan(t), 1/(t + 1) at t0 = 0 continuous at t0?

Determine the continuity of functions (e^t − 1)/t, tan(t), and 1/(t + 1) at t0 = 0 using limits and L'Hopital's Rule. Explore the behavior of these functions as t approaches 0, with step-by-step explanations. Ideal for students in Grades 11-12 studying calculus.

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