Question

In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { y \rightarrow - 3 } \frac { y ^ { 2 } + 4 y + 3 } { y ^ { 2 } - 3 }$$

Short answer

The limit of the function as \(y\) approaches -3 is 0.

Step-by-step solution

Substitute \(y = -3\) into the function

We start off by substituting \(y = -3\) into the function \(\frac { y ^ { 2 } + 4 y + 3 } { y ^ { 2 } - 3 }\).

Simplify the numerator

The numerator \(y ^ { 2 } + 4 y + 3\) becomes \((-3) ^ { 2 } + 4 (-3) + 3 = 9 - 12 + 3 = 0\).

Simplify the denominator

The denominator \(y ^ { 2 } - 3\) becomes \((-3) ^ { 2 } - 3 = 9 - 3 =6\).

Calculate the limit

The limit is the quotient of the numerator and the denominator. So, \(\lim _ { y \rightarrow - 3 } \frac { y ^ { 2 } + 4 y + 3 } { y ^ { 2 } - 3 } = \frac {0}{6} = 0\).

Key concepts

Calculus

Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. It's a tool for solving problems in physics, engineering, economics, and beyond, providing a framework for modeling systems where change is a key factor. Furthermore, calculus helps in understanding the behavior of functions, finding maximum and minimum values, and calculating the area under a curve.

At the heart of calculus are the concepts of differentiation and integration, which are ways to measure change and accumulation, respectively. To carry out these processes, we frequently use the limit of a function. By grasping these concepts, students can tackle a wide array of mathematical and real-world problems with precision and rigor.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. Limits can show the value that a function approaches as the input gets infinitely close to a point, even if the function itself doesn't actually reach that value. The notation \(\lim_{x \to a} f(x)\) represents the limit of \(f(x)\) as \(x\) approaches \(a\).

There's a common misconception that substitution always solves limit problems. However, that approach works only when the function is continuous at that point. If direct substitution leads to an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then further analysis or other techniques must be used to evaluate the limit. The textbook exercise showed a situation where direct substitution yielded a valid result, simplifying the limit evaluation process significantly.

Graphical Analysis of Limits

Graphical analysis of limits involves visually examining the behavior of a function as it approaches a certain input value on its graph. This technique is particularly useful when dealing with functions that are not easily simplified by algebraic manipulation. By graphing the function, one can observe whether the function approaches a specific value or if there's a discontinuity, such as a hole or vertical asymptote.

In the context of the provided exercise, graphing the function \(\frac{y^2 + 4y + 3}{y^2 - 3}\) and observing the behavior as \(y\) approaches \( -3\) supports the analytical solution. The graphic would show that as \(y\) gets closer to -3 from the left and the right, the value of the function approaches 0. This visual confirmation reinforces the limit calculation. Utilizing both, analytical and graphical methods yields a deeper understanding of limits and ensures the accuracy of the result.