use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before…
use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before limit properties can be applied. lim x→10 x² - 100 / x - 10 select the correct choice below and fill in any answer boxes in your choice. a. lim x→10 x² - 100 / x - 10 = (type an integer or a simplified fraction.) b. the limit does not exist
Answer
Explanation:
Step1: Factor the numerator
We know that $x^{2}-100=(x + 10)(x - 10)$ by the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$ where $a=x$ and $b = 10$. So the limit becomes $\lim_{x\rightarrow10}\frac{(x + 10)(x - 10)}{x - 10}$.
Step2: Simplify the expression
Since $x\neq10$ when taking the limit (we are approaching 10, not equal to 10), we can cancel out the common factor $(x - 10)$ in the numerator and the denominator. The expression simplifies to $\lim_{x\rightarrow10}(x + 10)$.
Step3: Apply the limit property
According to the limit property $\lim_{x\rightarrow a}(f(x)+g(x))=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow a}g(x)$, for $f(x)=x$ and $g(x)=10$, we have $\lim_{x\rightarrow10}(x + 10)=\lim_{x\rightarrow10}x+\lim_{x\rightarrow10}10$. And we know that $\lim_{x\rightarrow a}x=a$ and $\lim_{x\rightarrow a}c=c$ (where $c$ is a constant). So $\lim_{x\rightarrow10}x+\lim_{x\rightarrow10}10=10 + 10$.
Answer:
20