To find the limit [tex]\(\lim_{x \rightarrow \infty} 10^{5-x}\)[/tex], let's proceed through a detailed, step-by-step solution:
1. Rewrite the Exponential Expression: Start by observing the expression [tex]\(10^{5-x}\)[/tex]. This can be rewritten using exponent properties:
[tex]\[
10^{5-x} = 10^5 \cdot 10^{-x}
\][/tex]
2. Separate the Terms: Notice that [tex]\(10^5\)[/tex] is a constant (a fixed number) and it does not depend on [tex]\(x\)[/tex]. We can therefore separate the expression as follows:
[tex]\[
10^5 \cdot 10^{-x}
\][/tex]
3. Focus on [tex]\(10^{-x}\)[/tex]: Now, let's concentrate on the term [tex]\(10^{-x}\)[/tex]. As [tex]\(x\)[/tex] approaches infinity, the exponent [tex]\(-x\)[/tex] approaches negative infinity. Recall that as any positive base (in this case, 10) is raised to a negative exponent that grows larger and larger, its value gets closer and closer to 0.
4. Limit of [tex]\(10^{-x}\)[/tex] as [tex]\(x \rightarrow \infty\)[/tex]:
[tex]\[
\lim_{x \rightarrow \infty} 10^{-x} = 0
\][/tex]
5. Combine with Constant Multiplication: Now, we multiply [tex]\(10^5\)[/tex] by the result of the limit we found:
[tex]\[
\lim_{x \rightarrow \infty} 10^{5-x} = 10^5 \cdot \lim_{x \rightarrow \infty} 10^{-x} = 10^5 \cdot 0 = 0
\][/tex]
Therefore, the limit is:
[tex]\[
\lim_{x \rightarrow \infty} 10^{5-x} = 0
\][/tex]
