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The algebraic method is a way of finding the limit of a function by performing operations directly on the function's equation. To illustrate this, let's consider the exercise given: find the limit as x approaches 3 for the function 2x + 5.

Step by step:

• First, identify the limit expression, which in this case is \( \lim_{x \rightarrow 3}(2x+5) \)
• Next, substitute the value x is approaching (3) into the function: \(2(3)+5 \) This step is also known as 'direct substitution.'
• Finally, simplify the expression: \(2(3)+5 = 6+5=11\).
Thus, using the algebraic method, we find that \( \lim_{x \rightarrow 3}(2x+5)=11\).

Direct substitution is a straightforward method for finding limits. It involves substituting the value that x is approaching directly into the function. If the function is continuous at that point, you will get the limit result immediately.

For our example, the function is \(2x + 5\) and we are finding the limit as x approaches 3. By substituting 3 for x: \(2(3) + 5 \):

- Calculate \(2(3) = 6\)
- Then, add 5, giving us 11.

Thus, using direct substitution, we find that \(2(3) + 5 = 11\). This shows that \( \lim_{x \rightarrow 3}(2x+5) = 11\). It's a quick and easy way, especially for polynomials and other continuous functions where the operation can be done directly without complications.

Graphical verification can serve as a visual confirmation of the limit obtained. Let's visualize the function \(y = 2x + 5\)

• Draw the graph of this linear function, starting at the y-intercept (0, 5).
• Additionally, the slope is 2, meaning for every 1 unit increase in x, y increases by 2.

As x approaches 3, you can follow the graph towards x = 3 and observe the corresponding y-value. It should approach 11, confirming that the limit \( \lim_{x \rightarrow 3}(2x+5)\) is indeed 11.

This visual approach can help give a more intuitive understanding of limits and how the function behaves as x approaches a specific value.

Limits are foundational in calculus and serve as a cornerstone for defining derivatives and integrals. They describe the behavior of a function as the input approaches a specific value.

The notation \( \lim_{x \rightarrow a} f(x) \) expresses how f(x) behaves as x gets closer to a. If we say \( \lim_{x \rightarrow a} f(x) = L \), it means as x gets arbitrarily close to a, the function's value approaches L.

For instance, in the exercise given: \( \lim_{x \rightarrow 3}(2x+5) \)

• We are looking at how \(2x + 5\) behaves as x gets close to 3. Using algebra, we found the limit is 11.
• Direct substitution and graphical verification both confirmed this.

Understanding limits is crucial since they allow us to delve deeper into the behavior and properties of functions, paving the way for concepts like continuity, derivatives, and more advanced topics in calculus.