Solved on Mar 10, 2024

Find the limit of the function 2log8(3x)-2 \log_{8}(3-x) as xx approaches 3 from the left.

STEP 1

Assumptions
1. We are asked to find the limit of the function 2log8(3x)-2 \log _{8}(3-x) as xx approaches 33 from the left.
2. The logarithm is base 88.
3. The function is continuous at all points in its domain.

STEP 2

Understand the behavior of the function as xx approaches 33 from the left.
limx32log8(3x)\lim _{x \rightarrow 3-}-2 \log _{8}(3-x)

STEP 3

Recognize that as xx approaches 33 from the left, (3x)(3-x) approaches 00 from the right.

STEP 4

Since the logarithm function is undefined for non-positive values, we must ensure that (3x)(3-x) remains positive as xx approaches 33 from the left.

STEP 5

Rewrite the limit by substituting u=3xu = 3-x.
limx32log8(3x)=limu0+2log8(u)\lim _{x \rightarrow 3-}-2 \log _{8}(3-x) = \lim _{u \rightarrow 0+}-2 \log _{8}(u)

STEP 6

Now, we can see that as xx approaches 33 from the left, uu approaches 00 from the right.

STEP 7

Evaluate the limit of the logarithmic function as uu approaches 00 from the right.
limu0+log8(u)\lim _{u \rightarrow 0+}\log _{8}(u)

STEP 8

Recall that logb(x)\log _{b}(x) approaches -\infty as xx approaches 00 from the right for any base b>1b > 1.

STEP 9

Since 88 is greater than 11, the limit of log8(u)\log _{8}(u) as uu approaches 00 from the right is -\infty.
limu0+log8(u)=\lim _{u \rightarrow 0+}\log _{8}(u) = -\infty

STEP 10

Multiply the limit of the logarithmic function by 2-2.
limu0+2log8(u)=2×()\lim _{u \rightarrow 0+}-2 \log _{8}(u) = -2 \times (-\infty)

STEP 11

Recognize that multiplying a negative number by negative infinity results in positive infinity.
2×()=+-2 \times (-\infty) = +\infty

STEP 12

Conclude that the limit of the original function as xx approaches 33 from the left is positive infinity.
limx32log8(3x)=+\lim _{x \rightarrow 3-}-2 \log _{8}(3-x) = +\infty
The limit is ++\infty.

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