Divide numerator and denominator by the highest power of x in the denominator to find the limit.
-$\lim _{x \rightarrow\infty} \frac{5 x+3}{\sqrt{3 x^{2}+1}}$

A) $\frac { 5 } { \sqrt { 3 } }$
B) 0
C) $\frac { 5 } { 3 }$
D) $\infty$

Question

Divide numerator and denominator by the highest power of x in the denominator to find the limit.
-$\lim _{x \rightarrow\infty} \frac{5 x+3}{\sqrt{3 x^{2}+1}}$

A) $\frac { 5 } { \sqrt { 3 } }$
B) 0
C) $\frac { 5 } { 3 }$
D) $\infty$

Quizplus · Accepted Answer

The highest power of x in the denominator is x². Therefore, divide the numerator and denominator by x²:

\begin{align*}
\lim _{x ightarrow\infty} \frac{5 x+3}{\sqrt{3 x^{2}+1}} &= \lim _{x ightarrow\infty} \frac{5 + \frac{3}{x}}{\sqrt{3 + \frac{1}{x^{2}}}} &&	ext{divide numerator and denominator by }x^{2}\
&= \frac{5}{\sqrt{3}} \cdot \lim _{x ightarrow\infty} \frac{1}{\sqrt{1 + \frac{1}{3x^{2}}}} \
&= \frac{5}{\sqrt{3}} \cdot \frac{1}{\sqrt{1 + 0}} &&	ext{as}\ \lim _{x ightarrow\infty} \frac{1}{3x^{2}} =0\
&= \frac { 5 } { \sqrt { 3 } }
\end{align*}

Therefore, the limit is $\frac{5}{\sqrt{3}}$, which corresponds to choice A.

Divide Numerator and Denominator by the Highest Power of X lim⁡x→∞5x+33x2+1\lim _{x \rightarrow\infty} \frac{5 x+3}{\sqrt{3 x^{2}+1}}limx→∞​3x2+1​5x+3​

Divide Numerator and Denominator by the Highest Power of X $\lim _{x \rightarrow\infty} \frac{5 x+3}{\sqrt{3 x^{2}+1}}$