Question

Even function limits Suppose $f$ is an even function where $\underset{x\to 1^{-}}{lim}f(x)=5$ and $\underset{x\to 1^{+}}{lim}f(x)=6.$ Find $\underset{x\to -1^{-}}{lim}f(x)$ and $\underset{x\to -1^{+}}{lim}f(x)$.

Short answer

Answer: $\underset{x\to -1^{-}}{lim}f(x)=5$ and $\underset{x\to -1^{+}}{lim}f(x)=6$.

Step-by-step solution

Use the definition of even functions

Since $f$ is an even function, we can say that $f(-x)=f(x)$ for all $x$. This property will help us find the desired limits.

Find $\underset{x\to -1^{-}}{lim}f(x)$

To find $\underset{x\to -1^{-}}{lim}f(x)$, let's replace $x$ with $-x$. Then we get $\underset{-x\to -(-1{)}^{-}}{lim}f(-x)$. Using the definition of even functions, we rewrite the expression as $\underset{-x\to 1^{-}}{lim}f(x)$. We know this limit is equal to 5, so $\underset{x\to -1^{-}}{lim}f(x)=5$.

Find $\underset{x\to -1^{+}}{lim}f(x)$

Similarly, to find $\underset{x\to -1^{+}}{lim}f(x)$, we replace $x$ with $-x$. So, we have $\underset{-x\to -(-1{)}^{+}}{lim}f(-x)$. Now using even functions, we rewrite the expression as $\underset{-x\to 1^{+}}{lim}f(x)$. We know this limit is equal to 6. Therefore, $\underset{x\to -1^{+}}{lim}f(x)=6$. Hence, we found the two desired limits: $\underset{x\to -1^{-}}{lim}f(x)=5$ and $\underset{x\to -1^{+}}{lim}f(x)=6$.

Key concepts

Limit

A limit is a fundamental concept in calculus and analysis concerned with the behavior of a function as its argument approaches a particular value. When we talk about the limit of a function, we are asking what value the function is approaching as the input gets closer and closer to a certain point. For instance, in the exercise, the limit of the function as $x$ approaches 1 from different sides shows different values.The limit helps us understand the behavior of functions at specific points, even when the function is not explicitly defined there. Limits are foundational for defining derivatives and integrals, making them crucial for many practical applications. In understanding limits, it is important to examine the direction from which $x$ approaches a point because the output might vary depending on that direction. This brings us to one-sided limits, like left-hand and right-hand limits.

Left-hand Limit

The left-hand limit of a function as $x$ approaches a certain value examines the behavior of the function as $x$ approaches from the left side. It's signified with a minus ($-$) superscript, for instance, $x\to 1^{-}$.In the given exercise, the left-hand limit for $f(x)$ as $x$ approaches 1 is found to be 5. This means that as $x$ gets closer to 1 from values less than 1, the function value converges to 5.- **Why is it important?** - Helps in understanding how functions behave as they approach a certain point from one side. - Useful in piecewise functions and discontinuities where the function behavior differs depending on the direction. For even functions, these values at negative x-positions directly relate to the positive positions because $f(-x)=f(x)$. This symmetry allows us to determine the left-hand limits at $-1$ based on the information given at $1$.

Right-hand Limit

The right-hand limit is similar to the left-hand limit, but it involves approaching a certain value from the right side. This time, we use a plus ($+$) superscript, for example, $x\to 1^{+}$.In the context of the exercise, the right-hand limit for $f(x)$ as $x$ approaches 1 is 6, meaning as $x$ comes closer to 1 from values greater than 1, the output approaches 6.- **Key Details:** - Illustrates function behavior approaching from the positive side. - These limits help in assessing if a function is continuous by comparing with the left-hand limit at that point.For even functions, we use this symmetry to infer the right-hand limit for $-1$ from the known limit at $1$. The fact that $f(x)$ being even means its behavior at $x$ and $-x$ are linked, simplifies finding these limits.