A function f is even if f(−x)=f(x), for all x in the domain of...

Question

A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.

lim x→−2^− f(x)

Accepted Answer

Welcome back, everyone. Find the given limit for an even function F for all X in the domain of F if limit as X approaches 5 from the right of F of X is equal to 55, and we want to find the limit as X approaches -5 from the left of F of X. We're given four answer choices A says 0, B 55, C 55, and D-1. We're looking at an even function, so to visualize the steps, we're going to take one of the most primitive or elementary even functions, and that would be X 2 Y equals x2. That's one of the possible parabolas which Has origin as the point that it crosses, right? So essentially we have a parabola Y equals x2 that is an even function, and we have to recall that for even functions F of X is going to be equal to f of negative X, and this is exactly what we have. Let's suppose that we consider a point Y equals. Actually X equals -5. And then X equals 5. Based on this property. The value of the function. will be the same for -5 and 5 because the function is symmetrical relative to the y axis. Now, how do we apply limits in such a case? Well, essentially if we approach -5 from the left, and if we approach 5 from the right. The value of the limit is going to be the same as well. So we're going to define the property such that limit as X approaches some number negative A from the left of F of X. This limit must be equal to the limit as X approaches positive A from the right of the same function F of X, as long as F of X is even. And now using this property, we're given that limit as X approaches 5 from the right of F of X, this is equal to 55. If we use this property, then limit as X approaches. The, the, the opposite number to the previous number, so that's, that was 5, now we're taking -5, and we're taking the opposite sign instead of approaching it from the right, now we're approaching it from the left. This must be equal to 55 as well based on the property. So our final answer to this problem is going to be answer choice B 55. Thank you for watching.

A function f is even if f(−x)=f(x), for all x in the domain of f. Suppose f is even, with lim x→2^+ f(x)=5 and lim x→2^− f(x)=8. Evaluate the following limits.
lim x→−2^− f(x)

Key Concepts

Even Functions

One-Sided Limits

Limit Evaluation at Symmetric Points

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