Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. \(\lim _{t \rightarrow 3^{-}} \frac{t^{2}}{9-t^{2}}\)

Step-by-Step Solution

Answer

The limit is \( +\infty \).

Step 1: Identify the One-Sided Limit

The notation \( \lim_{t \rightarrow 3^{-}} f(t) \) indicates that we are looking for the left-hand limit as \( t \) approaches 3, which means values of \( t \) are slightly less than 3.

Step 2: Substitute the Limit Value in the Expression

Attempt to substitute \( t = 3 \) directly into the expression \( \frac{t^{2}}{9-t^{2}} \), so it becomes \( \frac{3^{2}}{9-3^{2}} = \frac{9}{9-9} = \frac{9}{0} \), which is undefined.

Step 3: Analyze the Behavior Around the Point

Since direct substitution leads to an undefined form, analyze the expression as \( t \to 3^- \): \( 9-t^2 = 9-(9-2\epsilon) = 0+2\epsilon \) which will approach 0 from the positive side as \( t \) approaches 3 from the left.

Step 4: Determine the Trend of the Expression

For \( t < 3 \), the denominator \( 9-t^2 \) becomes slightly positive, leading the fraction \( \frac{9}{0^+} \) towards positive infinity, thus indicating that the limit from the left ( \( t \to 3^- \) ) goes to positive infinity.

Step 5: Conclude the Limit

Therefore, the left-hand limit of the function \( \lim_{t \to 3^-} \frac{t^2}{9-t^2} \) is approaching positive infinity.

Key Concepts

One-sided limit

When dealing with limits, a one-sided limit refers to the value a function approaches as the input approaches a specified point from one side only. In the exercise, the notation \( \lim_{t \rightarrow 3^{-}} f(t) \) means we are investigating the behavior of the function as \( t \) draws closer to 3 from values less than 3. This is called the left-hand side approach because \( t \) comes from the left on the number line.
One-sided limits are useful to study points where a function may not be defined at the point itself from both sides. Understanding these limits can help in solving problems related to discontinuities, infinite limits, and analyzing behavior precisely.

Undefined expression

An undefined expression occurs in mathematics when you have a division by zero or an expression which does not have a finite or real number result. In this exercise, substituting \( t = 3 \) into the expression \( \frac{t^2}{9-t^2} \) results in division by zero: \( \frac{9}{0} \).
Zero in the denominator makes the expression undefined because division by zero is not possible in standard arithmetic. This means the value of the function at the point is not well determined, and other methods like limit evaluation are used to examine the behavior around this point.
Understanding undefined expression in terms of limits helps in using calculus tools to assess how functions behave near problematic points.

Behavior analysis

Behavior analysis involves examining how the function behaves as it approaches a certain point. Since our expression is undefined at the point, we need to look closely at how it behaves as \( t \) tends toward 3 from less than 3. For the expression \( \frac{t^{2}}{9-t^{2}} \), as \( t \) approaches 3 from the left, \( 9-t^2 \) becomes very small and positive, given by \( 0 + 2\epsilon \), where \( \epsilon \) is a very small positive number.
This behavior indicates that as \( t \) approaches 3, the denominator nears zero from the positive side. As a result, the entire fraction \( \frac{9}{0^+} \) trends towards positive infinity.
Behavior analysis in calculus provides insights into the tendencies of functions beyond just calculating limits, helping to understand the bigger picture around certain points.

Left-hand limit

A left-hand limit is concerned with approaching the limit value from the left, or from values slightly less than the target value. The notation \( \lim_{t \rightarrow 3^-} \) tells us to look at how the function performs when \( t \) is less than 3.In our example, as \( t \) approaches 3 from the left, the function \( \frac{t^2}{9-t^2} \) moves towards infinity. The denominator becomes very small and positive, leading the fraction to grow larger without bounds.
Evaluating left-hand limits allows us to fully understand scenarios where a function might exhibit different behavior right at the boundary compared to approaching it from the left.
This technique is particularly useful for identifying discontinuities, breaking down complex expressions, and predicting how functions deviate near singular points.