Question

If \(f\) is a continuous function such that \(\lim _{x \rightarrow \infty} f(x)=5,\) find, if possible, \(\lim _{x \rightarrow-\infty} f(x)\) for each specified condition. (a) The graph of \(f\) is symmetric to the \(y\) -axis. (b) The graph of \(f\) is symmetric to the origin.

Short answer

For symmetric functions across the y-axis, \( \lim_{{x \to - \infty}} f(x) = 5 \). While for symmetric functions across the origin, \( \lim_{{x \to - \infty}} f(x) = -5 \).

Step-by-step solution

Calculate limit for symmetry across y-axis

For a function to be symmetric around the y-axis, it means it follows the rule \(f(-x) = f(x)\). In such case, the limit as \(x\) approaches negative infinity should be similar to when \(x\) approaches positive infinity since the two sides mirror each other. Therefore, as \(x\) approaches negative infinity, the limit will also be 5 which becomes \( \lim_{{x \to -\infty}} f(x) = 5 \).

Calculate limit for symmetry across the origin

For a function to be symmetric around the origin, it implies that it obeys the rule \(f(-x) = -f(x)\). For this case, as \(x\) approaches positive infinity, we have the limit of \(f(x)\) as 5. Since the function is odd and thus changes sign, if we approach negative infinity, we should change the sign of the limit as well. Therefore, for \(f(-x)\), the limit will become negative 5, or \( \lim_{{x \to -\infty}} f(x) = -5 \).