Find the value of the limit $\lim _ { x \rightarrow 9 } \frac { \sqrt { x - 3 } } { x - 9 }$
A)-5
B)-3
C)-1
D)0
E)1
F)3
G)5
H) $\frac { 1 } { 6 }$

Question

Quizplus · Accepted Answer

To solve this limit, we can use algebraic manipulation to simplify the expression. The given limit is $\lim _ { x \rightarrow 9 } \frac { \sqrt { x - 3 } } { x - 9 }$. We can multiply the numerator and the denominator by the conjugate of the numerator, which is $\sqrt { x - 3 }$ + 3, to get rid of the square root in the denominator:$$\lim _ { x \rightarrow 9 } \frac { \sqrt { x - 3 } } { x - 9 } \cdot \frac { \sqrt { x - 3 } + 3 } { \sqrt { x - 3 } + 3 }$$This simplifies to:$$\lim _ { x \rightarrow 9 } \frac { x - 3 + 3\sqrt { x - 3 } + 9 } { (x - 9)(\sqrt { x - 3 } + 3) }$$Further simplification gives:$$\lim _ { x \rightarrow 9 } \frac { x + 3\sqrt { x - 3 } + 6 } { (x - 9)(\sqrt { x - 3 } + 3) }$$Notice that the original denominator was $x - 9$, and after multiplying by the conjugate, we have a factor of $x - 9$ in the denominator. The expression simplifies to:$$\lim _ { x \rightarrow 9 } \frac { 1 } { \sqrt { x - 3 } + 3 }$$Substituting $x = 9$ into the simplified expression:$$\frac { 1 } { \sqrt { 9 - 3 } + 3 } = \frac { 1 } { 3 + 3 } = \frac { 1 } { 6 }$$Therefore, the value of the limit is $\frac { 1 } { 6 }$.

Find the Value of the Limit lim⁡x→9x−3x−9\lim _ { x \rightarrow 9 } \frac { \sqrt { x - 3 } } { x - 9 }limx→9​x−9x−3​​

Find the Value of the Limit $\lim _ { x \rightarrow 9 } \frac { \sqrt { x - 3 } } { x - 9 }$