CALCU-6. Show that _(x, y) (-1.2)(y-2) /(x+1) doe

6. Show that

\lim _{(x, y) \rightarrow(-1.2)}(y-2) /(x+1)

does not exist.

Solution by Steps

step 1

To determine if the limit exists, we need to check if the limit is the same from all paths to the point

(-1,2)

step 2

The given limit from the asksia-ll calculator is incorrect because it suggests a numerical limit of 16, which does not consider the path-dependence of the limit in two variables

step 3

We will approach

(-1,2)

along different paths and see if the limits are the same. For example, along the line

y=x

, the expression becomes

\lim_{x \to -1} (x-2)/(x+1)

step 4

Simplifying the expression along

y=x

, we get

\lim_{x \to -1} (x-2)/(x+1) = (-1-2)/(-1+1) = -3/0

, which is undefined

step 5

Now, let's approach along the line

y=-x

, the expression becomes

\lim_{x \to -1} (-x-2)/(x+1)

step 6

Simplifying the expression along

y=-x

, we get

\lim_{x \to -1} (-x-2)/(x+1) = (1-2)/(-1+1) = -1/0

, which is also undefined

step 7

Since the limits along different paths are not the same (in fact, both are undefined but in different ways), the original limit does not exist

Answer

The limit

\lim _{(x, y) \rightarrow(-1,2)}(y-2) /(x+1)

does not exist.

Key Concept

Limits in multiple variables must be the same from all paths to exist.

Explanation

The limit of a function of two variables at a point is defined only if it is the same regardless of the path taken to approach that point. In this case, the limit along different paths yields different results, indicating that the limit does not exist.

7. Show that

\lim _{(x, y) \rightarrow+(0.0)} \arctan (0.2 x / y)

does not exist.

Solution by Steps

step 1

To determine if the limit exists, we must check if the limit is the same when approaching (0,0) from different paths

step 2

Let's approach (0,0) along the y-axis (x=0). The expression becomes

\lim_{y \to 0} \arctan(0) = 0

step 3

Now, let's approach (0,0) along the line y=x. The expression becomes

\lim_{x \to 0} \arctan(0.2) = \arctan(0.2)

step 4

Since the limits from step 2 and step 3 are different, the original limit does not exist

Answer

The limit

\lim_{(x, y) \to (0,0)} \arctan(0.2x/y)

does not exist.

Key Concept

Existence of a limit in multivariable calculus

Explanation

For a limit to exist in multivariable calculus, the limit must be the same regardless of the path taken to approach the point in question. In this case, different paths yield different limits, so the limit does not exist.

how to depend the limit does not exist

Generated Graph

Solution by Steps

step 1

To find the definite integral of

x^4

from

x = 3

to

x = 5

, we use the power rule for integration

step 2

The power rule for integration states that the integral of

x^n

with respect to

x

is

\frac{x^{n+1}}{n+1}

plus a constant of integration

step 3

Applying the power rule to

x^4

, we get

\frac{x^{5}}{5}

plus a constant

step 4

We evaluate this antiderivative from

x = 3

to

x = 5

. This means we calculate

\frac{5^{5}}{5} - \frac{3^{5}}{5}

step 5

Calculating the values, we get

\frac{3125}{5} - \frac{243}{5}

step 6

Simplifying the result, we obtain

\frac{3125 - 243}{5}

step 7

This simplifies to

\frac{2882}{5}

, which is equal to

576.4

Answer

\frac{2882}{5}

or

576.4

Key Concept

Definite Integration and Power Rule

Explanation

The definite integral of a power function

x^n

from

a

to

b

is found by evaluating the antiderivative

\frac{x^{n+1}}{n+1}

at the upper and lower limits of integration and subtracting the two results.

7. Show that

\lim _{(x, y) \rightarrow+(0.0)} \arctan (0.2 x / y)

does not exist.

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