Chapter 2: Problem 23
, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{t \rightarrow 1} \frac{t^{2}-1}{\sin (t-1)} $$
Short Answer
Expert verified
The limit is 2.
Step by step solution
01
Set Up the Limit Expression
The given limit expression is \( \lim _{t \rightarrow 1} \frac{t^{2}-1}{\sin (t-1)} \). This expression indicates we need to evaluate the behavior of this function as \( t \) approaches 1.
02
Simplify the Numerator
Notice that \( t^2 - 1 \) can be factored using the difference of squares formula: \( t^2 - 1 = (t - 1)(t + 1) \). Therefore, the expression becomes \( \lim _{t \rightarrow 1} \frac{(t-1)(t+1)}{\sin (t-1)} \).
03
Evaluate the Function Near the Limit Point
Substitute values near \( t = 1 \). You can use numerical values slightly greater and less than 1, like 0.99 and 1.01, to compute the value of \( \frac{t^{2}-1}{\sin (t-1)} \) using a calculator. Note the behavior of the results.
04
Use L'Hôpital's Rule
Recognize that directly substituting \( t = 1 \) gives a \( \frac{0}{0} \) indeterminate form. Apply L'Hôpital's Rule which requires differentiating the numerator and the denominator: \[ \lim_{t \rightarrow 1} \frac{2t}{\cos(t-1)}.\]
05
Compute the Limit Using L'Hôpital's Rule
Evaluate the derived expression \( \lim_{t \rightarrow 1} \frac{2t}{\cos(t-1)} \). Substitute \( t = 1 \) directly to obtain \( \frac{2 \, (1)}{\cos(0)} = \frac{2}{1} = 2 \).
06
Graph the Function Near the Limit Point
Use a graphing calculator or graphing software to plot the function \( \frac{t^{2}-1}{\sin (t-1)} \) around \( t = 1 \). Observe that as \( t \) approaches 1, the value of the function approaches 2.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphical Analysis
Graphical analysis is a powerful tool in calculus, especially when dealing with limits. To understand the function \( \frac{t^2 - 1}{\sin(t - 1)} \) as \( t \) approaches 1, we start by using a graphing calculator.
This will help us visualize what happens to the values of the function near the limit point. A graph provides a clear picture of the function's behavior and helps us confirm the analytical work we do.
By plotting the function:\[\frac{t^2 - 1}{\sin(t - 1)}\]on the graphing calculator around \( t = 1 \), we can see how the curve approaches a specific value. This approach helps in discerning trends and patterns that might not be evident through numerical values alone. Often, small graphs near the point can highlight any asymptotes or discontinuities, which are crucial in understanding the overall behavior of the function.
This will help us visualize what happens to the values of the function near the limit point. A graph provides a clear picture of the function's behavior and helps us confirm the analytical work we do.
By plotting the function:\[\frac{t^2 - 1}{\sin(t - 1)}\]on the graphing calculator around \( t = 1 \), we can see how the curve approaches a specific value. This approach helps in discerning trends and patterns that might not be evident through numerical values alone. Often, small graphs near the point can highlight any asymptotes or discontinuities, which are crucial in understanding the overall behavior of the function.
Indeterminate Forms
In calculus, indeterminate forms occur when substituting a value into an expression results in an undefined or undetermined value. The classic forms are \( \frac{0}{0} \) or \( \infty / \infty \).
Here, the expression \( \lim_{t \rightarrow 1} \frac{t^2 - 1}{\sin(t-1)} \) results in a \( \frac{0}{0} \) form when \( t = 1 \) is directly substituted.
Indeterminate forms like these require further manipulation to find the true limit. Without manipulation, such as factoring or using specialty rules like L'Hôpital's Rule, it is impossible to determine the actual limiting value. This step is crucial before applying any additional rules or calculations to resolve the problem completely.
Here, the expression \( \lim_{t \rightarrow 1} \frac{t^2 - 1}{\sin(t-1)} \) results in a \( \frac{0}{0} \) form when \( t = 1 \) is directly substituted.
Indeterminate forms like these require further manipulation to find the true limit. Without manipulation, such as factoring or using specialty rules like L'Hôpital's Rule, it is impossible to determine the actual limiting value. This step is crucial before applying any additional rules or calculations to resolve the problem completely.
L'Hôpital's Rule
Introduced to resolve indeterminate forms, L'Hôpital's Rule is a pivotal tool in calculus. It allows us to evaluate limits of indeterminate forms by differentiating the numerator and the denominator separately.
In this problem, substituting \( t = 1 \) into \( \frac{t^2 - 1}{\sin(t - 1)} \) results in \( \frac{0}{0} \). Applying L'Hôpital's Rule, we find the derivatives of the numerator \((2t)\) and the denominator \((\cos(t - 1))\).
This changes the expression to:\[\lim_{t \rightarrow 1} \frac{2t}{\cos(t-1)}\]By substituting \( t = 1 \) into this new expression, we quickly find the limit as 2. Thus, L'Hôpital's Rule offers a structured way to overcome challenges presented by indeterminate forms, ensuring correct results.
In this problem, substituting \( t = 1 \) into \( \frac{t^2 - 1}{\sin(t - 1)} \) results in \( \frac{0}{0} \). Applying L'Hôpital's Rule, we find the derivatives of the numerator \((2t)\) and the denominator \((\cos(t - 1))\).
This changes the expression to:\[\lim_{t \rightarrow 1} \frac{2t}{\cos(t-1)}\]By substituting \( t = 1 \) into this new expression, we quickly find the limit as 2. Thus, L'Hôpital's Rule offers a structured way to overcome challenges presented by indeterminate forms, ensuring correct results.
Numerical Approximation
Numerical approximation refers to estimating the value of a limit by using values close to the limit point. This is particularly useful when direct calculation is complex or when seeking confirmation of analytical methods.
In this exercise, values slightly below and above \( t = 1 \), such as 0.99 and 1.01, can be plugged into the function \( \frac{t^2 - 1}{\sin(t - 1)} \).
This provides an understanding of the function's behavior near \( t = 1 \), allowing us to approximate the limit.
Using technological tools, like a calculator, aids this approximation process. Though not as precise as solving analytically, numerical approximations serve well as a double-check or quick estimation, validating the expected behavior as the function approaches the limit point.
In this exercise, values slightly below and above \( t = 1 \), such as 0.99 and 1.01, can be plugged into the function \( \frac{t^2 - 1}{\sin(t - 1)} \).
This provides an understanding of the function's behavior near \( t = 1 \), allowing us to approximate the limit.
Using technological tools, like a calculator, aids this approximation process. Though not as precise as solving analytically, numerical approximations serve well as a double-check or quick estimation, validating the expected behavior as the function approaches the limit point.
