find-the-limits-in-exercises-1-12-lim-_x-y-rightarrow00-fracey-sin-xx-3

Question

Find the limits in Exercises $1-12$.
$$
\lim _{(x, y) \rightarrow(0,0)} \frac{e^{y} \sin x}{x}
$$

Stephen Hobbs · Accepted Answer

and this problem will start off with evaluating e to the X and using direct substitution will get eat 20 multiplied by sine of X, divided by acts. And in this case, we have before we even substitute. We could imagine as X is approaching zero here. There&#x27;s a rule that tells us that this especially if we use Low Patel it would be co sign of X over one and then substituting Zero would just get us one. So really, we could use an identity here. You to the zero would also be one which would evaluate this whole limit, so one.

Stephen Hobbs · Answer

evaluating this limit for X approaches. Zero. Why approaches Elena too? We could use direct substitution, so we hav e to the zero or x minus Ln of to And then another way to rewrite this is we could think of it as e to the the zero goes away. And we can move whatever constant is in front of the natural log to the power. And so we could really imagine it as Ln of two to the power negative one. And then from here, we know that the, um, exponential and natural logger and verses of one another. And so our final solution would be to the negative one or one.

Zachary Mitchell · Answer

we have the limit as X approaches zero of negative one. Divide by X squared X plus one. Now in the denominator, we have an X squared that X squared is approaching zero and it&#x27;s always greater than or equal to zero as X approaches zero and notice that X plus one is is approaching one as X approaches. Zero. So ultimately we have that the denominator is approaching zero and is always greater than or equal to zero as X approaches zero and the numerator is a negative number. So we have negative. Infinity is the limit.

Zachary Mitchell · Answer

we have the limit as X goes to zero from the left hand side of 5/2 X. That means that the denominator is becoming very small and as it&#x27;s becoming very small, it&#x27;s nay. It has a negative value, so this means that the fraction is tending to negative infinity.

Zachary Mitchell · Answer

we have the limit limit as X goes to zero from the positive side of 1/3 X. So this means that the denominator here is getting very small, but it&#x27;s still positive. So one over a very small positive number tends to infinity. That&#x27;s it.

Zachary Mitchell · Answer

we have the limits as X approaches zero of one over X to the to thirds. Now notice that as X approaches zero, the denominator here approaches zero and the denominator is always greater than or equal to zero because the exponents is two thirds. Therefore, this limit is equal to infinity.

Problem

Find the limits in Exercises $1-12$. $$ \lim _{…

Problem 9 Easy Difficulty

Find the limits in Exercises $1-12$.
$$
\lim _{(x, y) \rightarrow(0,0)} \frac{e^{y} \sin x}{x}
$$

Answer

1

Related Courses

Thomas Calculus

Related Topics

Discussion

Top Calculus 3 Educators

Catherine R.

Anna Marie V.

Caleb E.

Michael J.

Calculus 3 Courses

Vectors Intro

Vector Basics - Example 1

Recommended Videos

Watch More Solved Questions in Chapter 14

Video Transcript

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano

Thomas Calculus

Catherine R.

Anna Marie V.

Caleb E.

Michael J.

Vectors Intro

Vector Basics - Example 1

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

1

Thomas Calculus

Catherine R.

Anna Marie V.

Caleb E.

Michael J.

Vectors Intro

Vector Basics - Example 1

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. GiordanoThomas Calculus

Catherine R.

Anna Marie V.

Caleb E.

Michael J.

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano

Thomas Calculus