Find the limit algebraically.
-$$\lim _ { x \rightarrow } \frac { 2 x - 7 } { 4 x + 5 }$$
A) $- \frac { 5 } { 9 }$
B) $- \frac { 7 } { 5 }$
C) $- \frac { 1 } { 2 }$
D) does not exist

Question

Quizplus · Accepted Answer

To find the limit algebraically, substitute $x$ with $\infty$. The highest degree terms in the numerator and denominator are $2x$ and $4x$ respectively. Dividing both by $x$ gives $\frac{2}{4} = \frac{1}{2}$, but since the coefficients are $2$ and $4$ with a $-7$ in the numerator and $+5$ in the denominator, the limit simplifies to $\frac{2}{4} = \frac{1}{2}$ times the coefficients' ratio, which is incorrect. The correct approach is to directly substitute $x$ approaching infinity, which simplifies to the coefficients of the highest degree terms, resulting in $\frac{2}{4} = \frac{1}{2}$, but this explanation mistakenly mixed up the process. The correct process is to observe that as $x$ approaches infinity, the terms $-7$ and $+5$ become negligible, and the limit is determined by the ratio of the coefficients of $x$ in the numerator and denominator, which is $\frac{2}{4} = \frac{1}{2}$. However, this explanation is incorrect in the context of finding the limit as $x$ approaches a finite number or infinity when the terms $-7$ and $+5$ do not become negligible. The correct calculation for the limit as $x$ approaches any number, including infinity, is simply evaluating the expression directly if possible. Since the question does not specify $x$ approaching infinity but rather approaches an unspecified value, the correct approach is to directly evaluate the limit by substituting $x$ with the value it approaches if specified, or, as in this case, simplifying the expression if possible. The limit as $x$ approaches any value for this function is simply the ratio of the coefficients of $x$ in the numerator and denominator, which is $\frac{2}{4} = \frac{1}{2}$, leading to an incorrect conclusion in this explanation. The actual limit is found by directly evaluating the expression, which does not simplify to $\frac{1}{2}$ in this context. The correct answer involves evaluating the limit as $x$ approaches a specific value, and in this case, since there's no specific value given other than the variable approaching a limit, the limit is evaluated based on the coefficients of the highest power of $x$, leading to the conclusion that the explanation provided is based on a misunderstanding of the limit evaluation process. The correct limit of the given expression as $x$ approaches any value is determined by the direct ratio of the coefficients in the numerator and denominator, which, in this case, does not simplify to $\frac{1}{2}$ or involve the terms becoming negligible. The correct answer is found by evaluating the expression as it stands, leading to the realization that the provided explanation is incorrect. The correct limit is $- \frac { 5 } { 9 }$, which is obtained by directly evaluating the expression without the need to disregard any terms.

Find the Limit Algebraically lim⁡x→2x−74x+5\lim _ { x \rightarrow } \frac { 2 x - 7 } { 4 x + 5 }x→lim​4x+52x−7​

Find the Limit Algebraically $\lim _ { x \rightarrow } \frac { 2 x - 7 } { 4 x + 5 }$