Question

Writing to Learn Let \(L\) be a real number, lim \(_{x \rightarrow c} f(x)=L\) and \(\lim _{x \rightarrow c} g(x)=\infty\) or \(-\infty .\) Can \(\lim _{x \rightarrow c}(f(x)+g(x))\) be determined? Explain.

Short answer

The limit \(\lim _{x \rightarrow c}(f(x)+g(x))\) cannot be determined as a finite real number. It will either be infinity or negative infinity.

Step-by-step solution

Understanding the problem

Here, we are given \(\lim _{x \rightarrow c} f(x)=L\) and \(\lim _{x \rightarrow c} g(x)=\infty\) or \(-\infty\). We are asked to determine the sum of these limits, or in other words, \(\lim _{x \rightarrow c}(f(x)+g(x))\) . Also, remember \(L\) is a real number.

Determine the sum of the limits

Generally, if the limit of a function as \(x\) approaches a certain value is infinity or negative infinity, this denotes the function's behavior of increasing or decreasing without bound respectively, as \(x\) approaches that certain value. When we add a finite number (in our case \(L\), which is a real number) to infinity or negative infinity, the sum is still infinity or negative infinity respectively. The presence of the finite number does not change the result.

Conclusion

So, in our problem, \(\lim _{x \rightarrow c}(f(x)+g(x))\) cannot be determined as a finite real number, because adding a finite real number to infinity or negative infinity still results in infinity or negative infinity respectively.