Question

Determine the following limits and justify your answers. $$\lim _{t \rightarrow 2} \frac{t^{2}+5}{1+\sqrt{t^{2}+5}}$$

Short answer

Answer: The limit is $\frac{9}{4}$.

Step-by-step solution

Substitute t = 2 directly

First, let's plug in t = 2 into the given expression and see what it evaluates to: $$\frac{2^2 + 5}{1+\sqrt{2^2 + 5}} = \frac{4 + 5}{1+\sqrt{4+5}} = \frac{9}{1+\sqrt{9}}$$ Since the expression evaluates to a defined value and there is no indeterminate form, we can directly find the limit.

Evaluate the limit

As t approaches 2, the expression evaluates to: $$\lim_{t\to 2} \frac{t^2+5}{1+\sqrt{t^2+5}} = \frac{9}{1+\sqrt{9}} = \frac{9}{1+3} = \frac{9}{4}$$ Thus, the limit is: $$\lim _{t \rightarrow 2} \frac{t^{2}+5}{1+\sqrt{t^{2}+5}}=\frac{9}{4}$$

Key concepts

Substitution method

Substitution is a fundamental concept in calculus, especially when solving limits. It involves replacing the variable in an expression with a specific value to evaluate a limit at that point.

Let's consider the expression \( \frac{t^{2}+5}{1+\sqrt{t^{2}+5}} \). The core idea here is to substitute \( t = 2 \) directly into the expression. By performing this substitution, we get:

• The expression becomes \( \frac{2^2 + 5}{1+\sqrt{2^2 + 5}} \)
• This simplifies to \( \frac{4 + 5}{1+\sqrt{9}} \)
• Finally, it results in \( \frac{9}{1+3} \)

The substitution method is straightforward and effective, provided that it does not lead to an indeterminate form such as \( \frac{0}{0} \). Thus, it's important to check if the substitution results in a well-defined expression, as it does here, pointing us directly to the solution.

Limit evaluation

Limit evaluation is the process of finding the value that a function approaches as the input of the function approaches some value. This often involves direct substitution, as seen in the exercise provided.

In the given problem, the direct substitution of \( t = 2 \) into the function \( \frac{t^{2}+5}{1+\sqrt{t^{2}+5}} \) yields a numerical value without complication. The expression simplifies as follows:

• Start with \( t = 2 \)
• Solve \( \frac{2^2 + 5}{1+\sqrt{4+5}} \)
• The limit becomes \( \frac{9}{4} \)

Evaluating the limit directly from these calculations makes it evident that the function behaves nicely around \( t = 2 \), thus achieving \( \lim _{t \rightarrow 2} \frac{t^{2}+5}{1+\sqrt{t^{2}+5}}=\frac{9}{4} \). Successful limit evaluation often hinges on the absence of indeterminate forms.

Indeterminate forms

Indeterminate forms occur in calculus when substituting a limit results in expressions like \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), or \( 1^\infty \), among others. These forms signal that standard substitution won't work, and more sophisticated techniques may be required.

In our given problem, attempting to substitute \( t \) directly into the function \( \frac{t^{2}+5}{1+\sqrt{t^{2}+5}} \) does not result in an indeterminate form. However, it's crucial to know when an expression does become indeterminate, as other methods such as L'Hopital's Rule or algebraic manipulation might be necessary.

Here are some common techniques to resolve indeterminate forms:

• Algebraic simplification: Factor or expand expressions to cancel out terms causing the indeterminate form.
• L'Hopital's Rule: Differentiate the numerator and denominator when faced with \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) forms.
• Rationalize expressions: Especially useful with square roots or other complex radicals.

Understanding these concepts not only helps in finding limits accurately but also expands your toolbox for tackling complex calculus problems.