Algebra of Limits - Practice Questions & MCQ

Algebra of Limits

Let $f(x)$ and $g(x)$ be defined for all $x \neq$ a over some open interval containing a. Assume that $L$ and $M$ are real numbers such that $\lim _{x \rightarrow a} f(x)=L$ and $\lim _{x \rightarrow a} g(x)=M$. Let c be a constant. Then, each of the following statements holds:

Sum law for limits : $\lim _{x \rightarrow a}(f(x)+g(x))=\lim _{x \rightarrow a} f(x)+\lim _{x \rightarrow a} g(x)=L+M$
Difference law for limits : $\lim _{x \rightarrow a}(f(x)-g(x))=\lim _{x \rightarrow a} f(x)-\lim _{x \rightarrow a} g(x)=L-M$
Constant multiple law for $\operatorname{limits}^{\lim } \lim _{x \rightarrow a} c f(x)=c \cdot \lim _{x \rightarrow a} f(x)=c L$
Product law for limits : $\lim _{x \rightarrow a}(f(x) \cdot g(x))=\lim _{x \rightarrow a} f(x) \cdot \lim _{x \rightarrow a} g(x)=L \cdot M$

Quotient law for limits: $\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}=\frac{L}{M}$ for $M \neq 0$
Power law for limits : $\lim _{x \rightarrow a}(f(x))^n=\left(\lim _{x \rightarrow a} f(x)\right)^n=L^n$ for every positive integer $n$ Composition law of limit: $\lim _{x \rightarrow a}(f o g)(x)=f\left(\lim _{x \rightarrow a} g(x)\right)=f(M)$, only if $\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{g}(\mathrm{x})=\mathrm{M}$.

If $\lim _{x \rightarrow a} f(x)=+\infty$ or $-\infty$, then $\lim _{x \rightarrow a} \frac{1}{f(x)}=0$