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QuestionFind lim⁡x→0f(x+0i)\lim _{x \rightarrow 0} f(x+0 i) and lim⁡y→0f(0+iy)\lim _{y \rightarrow 0} f(0+i y) for f(z)=z+zˉzf(z)=\frac{z+\bar{z}}{z}.

Studdy Solution

STEP 1

Assumptions1. The function is f(z)=z+zˉzf(z)=\frac{z+\bar{z}}{z} . We are asked to find the limits as x→0x \rightarrow0 for f(x+0i)f(x+0 i) and as y→0y \rightarrow0 for f(0+iy)f(0+i y)3. We are also asked to find the limit as z→0z \rightarrow0 for f(z)f(z)4. Here, zz is a complex number and zˉ\bar{z} is its conjugate

STEP 2

First, we need to find the limit as x→0x \rightarrow0 for f(x+0i)f(x+0 i). We can do this by substituting x+0ix+0 i into the function f(z)f(z).
f(x+0i)=x+0i+x+0i‾x+0if(x+0 i)=\frac{x+0 i+\overline{x+0 i}}{x+0 i}

STEP 3

Now, simplify the expression. The conjugate of a real number is the number itself.
f(x+0i)=x+xxf(x+0 i)=\frac{x+x}{x}

STEP 4

implify the expression further.
f(x+0i)=2xxf(x+0 i)=\frac{2x}{x}

STEP 5

Now, calculate the limit as x→0x \rightarrow0.
lim⁡x→0f(x+0i)=lim⁡x→02xx\lim{x \rightarrow0} f(x+0 i)=\lim{x \rightarrow0} \frac{2x}{x}

STEP 6

implify the limit expression.
lim⁡x→0f(x+0i)=2\lim{x \rightarrow0} f(x+0 i)=2

STEP 7

Next, we need to find the limit as y→0y \rightarrow0 for f(0+iy)f(0+i y). We can do this by substituting 0+iy0+i y into the function f(z)f(z).
f(0+iy)=0+iy+0+iy‾0+iyf(0+i y)=\frac{0+i y+\overline{0+i y}}{0+i y}

STEP 8

Now, simplify the expression. The conjugate of an imaginary number is the negative of the number.
f(0+iy)=0+iy+0−iy0+iyf(0+i y)=\frac{0+i y+0-i y}{0+i y}

STEP 9

implify the expression further.
f(+iy)=+iyf(+i y)=\frac{}{+i y}

STEP 10

Now, calculate the limit as y→0y \rightarrow0.
lim⁡y→0f(0+iy)=lim⁡y→000+iy\lim{y \rightarrow0} f(0+i y)=\lim{y \rightarrow0} \frac{0}{0+i y}

STEP 11

implify the limit expression.
lim⁡y→0f(0+iy)=0\lim{y \rightarrow0} f(0+i y)=0

STEP 12

Finally, we need to find the limit as z→0z \rightarrow0 for f(z)f(z). We can do this by substituting 00 into the function f(z)f(z).
f(0)=0+0‾0f(0)=\frac{0+\overline{0}}{0}

STEP 13

Now, simplify the expression. The conjugate of zero is zero.
f(0)=0+00f(0)=\frac{0+0}{0}

STEP 14

implify the expression further.
f(0)=00f(0)=\frac{0}{0}

STEP 15

The expression 00\frac{0}{0} is an indeterminate form. However, since the limits as x→0x \rightarrow0 and y→0y \rightarrow0 exist and are different, the limit as z→0z \rightarrow0 does not exist.
Therefore, lim⁡z→0f(z)\lim{z \rightarrow0} f(z) does not exist.

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