Question

Explain the meaning of \(\lim _{x \rightarrow-\infty} f(x)=10\).

Short answer

Answer: The given limit expression \(\lim_{x \rightarrow -\infty} f(x) = 10\) means that as the input x approaches negative infinity (i.e., becomes increasingly large in the negative direction), the function f(x) approaches a constant value of 10. This can be visualized on the graph of f(x) with a horizontal asymptote at y = 10.

Step-by-step solution

Understand the concept of limit

A limit is a value that a function approaches as the input (in this case, x) approaches a specified value. In our exercise, the input x is approaching negative infinity, which means that the function f(x) is being evaluated for increasingly large negative x values.

Interpret the given limit expression

The expression \(\lim_{x \rightarrow -\infty} f(x) = 10\) tells us that as the input x approaches negative infinity, the function f(x) approaches a constant value of 10. It's important to note that this does not mean that f(x) equals 10 for all negative values of x, but rather that the values of f(x) get closer and closer to 10 as x becomes larger and larger in the negative direction.

Visualize the limit on the graph of f(x)

To better understand the meaning of the limit, we can imagine the graph of the function f(x). Since the limit is 10 as x approaches negative infinity, the graph of f(x) would appear to get closer and closer to the horizontal line y = 10 as we move towards the left (in the negative x direction) on the graph. This horizontal line is called an asymptote, which represents a line that the function approaches but may never actually reach exactly.

Conclude the explanation

In conclusion, the expression \(\lim_{x \rightarrow -\infty} f(x) = 10\) means that as the input x gets increasingly large in the negative direction, the function f(x) approaches a constant value of 10. This can be visualized on the graph of the function as an asymptote at y = 10.