Question

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow-3}\left(2 x^{2}+4 x+1\right) $$

Short answer

The limit of the function \(2x^2+4x+1\) as \(x\) approaches -3 is 7.

Step-by-step solution

Identify the Function and Limiting Value

In this exercise, the function given is \(2x^2+4x+1\), and the limiting value of \(x\) is -3. We are asked to find \(\lim _{x \rightarrow-3}\left(2 x^{2}+4 x+1\right)\) .

Substitute the Limiting Value

As there is no clause that prohibits the direct substitution of -3 into this function, substitute \(x = -3\) into the function \(2x^2+4x+1\). When we substitute \(x = -3\) into the function, the expression becomes \(2(-3)^2 + 4(-3) + 1\).

Evaluate the Function

Simplify the expression \(2(-3)^2 + 4(-3) + 1\). Performing the operations, the expression becomes \(2(9) -12 + 1 = 18 - 12 + 1 = 7\)

Key concepts

Limit of a Function

Understanding the limit of a function is crucial to calculus and involves approaching a certain point, not necessarily reaching it. It essentially tells us the value that a function approaches as the input (or 'x' value) gets closer and closer to some number. In the context of our exercise, \(\lim _{x \rightarrow-3}\left(2 x^{2}+4 x+1\right)\), the limit describes what value the function \(2x^{2}+4x+1\) approaches as x approaches -3.

The beauty of limits lies in their ability to deal with situations where direct function evaluation isn't possible, like at an indeterminate form or where there's a discontinuity. In this case, there's no such issue, so we can expect the function to behave nicely as x nears -3.

Direct Substitution Method

The direct substitution method is a straightforward technique used in calculus to find the limit of a function when x approaches a particular value. It's the first method you should try, and it simply involves putting the value that x is approaching into the function itself. This is often the quickest way to evaluate limits when the function is continuous at the point x is approaching.

To apply this in our example, we directly substitute x with -3 in the function \(2x^2+4x+1\) because the function is defined at that point, which means there's no break or hole in the graph at \(x = -3\). If the function had a point of discontinuity at \(x = -3\), or if substituting -3 resulted in an undefined expression like division by zero, then we would have to use another method to find the limit.

Evaluating Functions

Evaluating functions involves calculating the value of a function for a specific x value. This task becomes the core of finding limits by direct substitution, where we replace the variable with the number we're approaching. For the given function in our exercise \(2x^2+4x+1\), the evaluation is straightforward.

After substituting \(x = -3\), you carry out basic algebra: calculate the square of -3, which is 9, multiply by 2 to get 18, and then calculate the rest of the expression to arrive at the final answer of 7. It's a simple plug-and-play approach, where arithmetic takes the front stage. It requires careful, step-by-step execution of operations, ensuring that mistakes in basic algebra don't lead to incorrect limit evaluations.