Question

Other techniques Evaluate the following limits, where a and \(b\) are fixed real numbers. $$\lim _{t \rightarrow 2} \frac{3 t^{2}-7 t+2}{2-t}$$

Short answer

Question: Find the limit of the rational function as t approaches 2: \(\lim _{t \rightarrow 2} \frac{3 t^{2}-7 t+2}{2-t}\) Answer: The limit of the given function as \(t \rightarrow 2\) is -5.

Step-by-step solution

Try direct substitution

Let's try plugging in t=2 into the given function: $$\frac{3(2)^2 - 7(2) + 2}{2 - 2} = \frac{12 - 14 + 2}{0} = \frac{0}{0}$$ We get an indeterminate form \(\frac{0}{0}\), so we can't evaluate the limit directly. We must find another method to simplify the function.

Factor the numerator

Observe that the numerator is a quadratic polynomial. Let's try factoring it: \(3t^2 - 7t + 2 = (3t - 1)(t - 2)\). Now, the function becomes: $$\lim _{t \rightarrow 2} \frac{(3 t - 1)(t - 2)}{2 - t}$$

Simplify the function

Notice that we can simplify the function by factoring out a negative sign from the denominator and then cancelling out the common term \((t-2)\): $$\lim _{t \rightarrow 2} \frac{(3 t - 1)(t - 2)}{-(t - 2)} = \lim _{t \rightarrow 2} \frac{(3 t - 1)(t - 2)}{-(t - 2)} \cdot \frac{-1}{-1} = \lim _{t \rightarrow 2} \frac{(3 t - 1)}{-1}$$ Now, the function is in a simpler form.

Compute the limit

Now that our function has been simplified, we can compute the limit by direct substitution: $$\lim _{t \rightarrow 2} \frac{(3 t - 1)}{-1} = \frac{3(2) - 1}{-1} = \frac{5}{-1} = -5$$ So, the limit of the given function as \(t \rightarrow 2\) is -5.

Key concepts

Indeterminate Forms

When evaluating limits, we often encounter expressions that result in undefined values, such as \(\frac{0}{0}\). These are known as indeterminate forms. Indeterminate forms can appear when substituting values directly into a function creates a division by zero or an undefined operation.

These situations require additional techniques to resolve. For example, simplifying the function into a different form through algebraic manipulation or utilizing mathematical theorems. Understanding indeterminate forms is crucial in calculus as it helps identify when a limit can be evaluated without direct substitution.

• Indeterminate forms signal a need for further analysis.
• Simplifying helps reveal the true behavior of the function.
• They often involve algebraic manipulation to find limits.

Direct Substitution

Direct substitution is a straightforward method for evaluating limits. You simply substitute the value towards which the variable is approaching into the function. This technique works well when the function is continuous and well-defined at that point.

However, if direct substitution results in an undefined expression, like \(\frac{0}{0}\), it indicates an indeterminate form, and another method must be used. This straightforward approach makes learning calculus easier, as it often provides quick answers for well-behaved functions.

• Direct substitution works when a function is continuous.
• Use it as the initial attempt to evaluate limits.
• Indeterminate results show that direct substitution isn't sufficient.

Factoring Polynomials

Factoring polynomials is a helpful method to simplify expressions, particularly when dealing with quadratic terms in limits. By expressing a polynomial as a product of simpler polynomials, you can often cancel out terms across numerators and denominators, allowing for more straightforward evaluation.

In this context, it's particularly useful for cases where direct substitution results in indeterminate forms. The act of factoring breaks down complex expressions, turning them into more manageable parts that can reveal hidden simplicity.

• Factoring simplifies expressions by breaking them down.
• It helps solve limit problems involving polynomials.
• Useful to eliminate zero terms in the numerator and denominator.

Quadratic Polynomials

Quadratic polynomials are expressions of the form \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are constants. Understanding how to work with these is essential as they frequently occur in limit problems.

When dealing with limits, the roots of quadratic polynomials can be useful in simplifying expressions, especially when factored. For example, the numerator of a rational expression can often be factored into a product of linear terms, thus providing opportunities to simplify if shared by the denominator.

• Quadratic polynomials are essential in algebra and calculus.
• They are frequently used in limits when factored.
• Factoring helps in revealing simplification opportunities.