Squeeze Theorem Multivariable- Practice Problems and Solutions

What Is the Squeeze Theorem in Multivariable Calculus?

The squeeze theorem (also called the sandwich theorem) lets you find the limit of a function by trapping it between two simpler functions whose limits you already know. If those two bounding functions converge to the same value, your target function gets squeezed into that same limit.

The formal statement looks like this:

If g(x,y) ≤ f(x,y) ≤ h(x,y) near (a,b), and lim g(x,y) = L = lim h(x,y), then lim f(x,y) = L.

The same principle applies to limits along paths, polar coordinates, and vector-valued functions. The multivariable version is straightforward once you understand the one-variable version.

Why Students Struggle With This Theorem

Most students fail the squeeze theorem not because the math is hard, but because they pick the wrong bounding functions. You need two functions that bracket your target function and are easier to evaluate at the limit point.

Another common mistake is forgetting that the squeeze theorem requires equality at the limit between the bounding functions. If the upper and lower bounds converge to different values, you get nothing.

Practice Problems and Solutions

Problem 1: Basic Two-Variable Squeeze

Evaluate: lim(x,y)→(0,0) x²y · sin(1/(xy))

This looks terrifying. The sin term oscillates wildly as (x,y) approaches (0,0). But notice that sin(1/(xy)) is bounded between -1 and 1.

Step 1: Identify the bounded function

For all real numbers: -1 ≤ sin(θ) ≤ 1

Step 2: Set up the squeeze

-x²|y| ≤ x²y · sin(1/(xy)) ≤ x²|y|

Step 3: Evaluate the bounds

lim(x,y)→(0,0) x²|y| = 0

Answer: The limit is 0.

Problem 2: Path-Dependent Limit with Squeeze

Evaluate: lim(x,y)→(0,0) (x²y)/(x⁴ + y²)

This limit does not exist along all paths. Let's check a couple.

Along y = x²: (x² · x²)/(x⁴ + x⁴) = x⁴/(2x⁴) = 1/2

Along y = x: (x² · x)/(x⁴ + x²) = x³/(x⁴ + x²) = x/(x² + 1) → 0

Different paths give different results, so the limit does not exist. The squeeze theorem doesn't apply here because no single value squeezes the function everywhere near (0,0).

Problem 3: Polar Coordinate Squeeze

Evaluate: lim(x,y)→(0,0) (x² + y²) · cos(x² + y²)

Step 1: Convert to polar coordinates

x² + y² = r², so we have r² · cos(r²)

Step 2: Apply the squeeze

Since -1 ≤ cos(r²) ≤ 1, we get:

-r² ≤ r² · cos(r²) ≤ r²

Step 3: Evaluate

lim(r→0) r² = 0

Answer: The limit is 0.

Problem 4: Squeeze with Exponential Bounds

Evaluate: lim(x,y)→(0,0) e^(-1/(x²+y²)) · sin(x)

Step 1: Recognize the squeeze opportunity

The exponential term e^(-1/(x²+y²)) is bounded between 0 and 1 for all (x,y) ≠ (0,0). As (x,y) → (0,0), this term approaches 0 extremely fast.

Step 2: Set up the bounds

-e^(-1/(x²+y²)) ≤ e^(-1/(x²+y²)) · sin(x) ≤ e^(-1/(x²+y²))

Step 3: Evaluate

lim(x,y)→(0,0) e^(-1/(x²+y²)) = 0

Answer: The limit is 0.

Problem 5: Vector-Valued Function Squeeze

Evaluate: lim(t→0) ⟨t · sin(1/t), t² · cos(1/t²)⟩

For vector-valued functions, apply the squeeze theorem to each component separately.

First component:

|t · sin(1/t)| ≤ |t|, and lim(t→0) |t| = 0

Second component:

|t² · cos(1/t²)| ≤ t², and lim(t→0) t² = 0

Answer: The limit is ⟨0, 0⟩.

Quick Reference: Common Bounding Strategies

How to Actually Use the Squeeze Theorem

Here's the step-by-step process you should follow every time:

• Step 1: Identify the "messy" part of your function that oscillates or behaves badly near your limit point
• Step 2: Find the maximum and minimum values of that messy part (usually -1 and 1 for trig functions)
• Step 3: Multiply those bounds by the rest of your function
• Step 4: Evaluate the limits of both bounding functions
• Step 5: If both limits equal the same value L, your function's limit is L

Most problems you'll encounter use one of three bounding functions: |f(x,y)| ≤ M (trig functions), 0 ≤ f(x,y) ≤ something (polynomials), or radical bounds like √(x²+y²) ≥ |x|.

Common Mistakes That Cost You Points

• Forgetting to check all paths first. If a limit doesn't exist along different paths, the squeeze theorem won't save you. Always verify existence before applying squeeze.
• Setting up wrong inequalities. Your bounding functions must actually bracket the original function near the limit point, not just at the point itself.
• Assuming symmetry. Just because something works in one quadrant doesn't mean it works everywhere. Check the full neighborhood.
• Ignoring domain restrictions. The squeeze theorem requires the function to be defined near (not at) the limit point. Check your domain first.

When Squeeze Theorem Is the Wrong Tool

The squeeze theorem only works when your function is bounded by two functions that converge to the same limit. If the bounds converge to different values, you get no information. If your function isn't bounded near the limit point, squeeze doesn't apply.

For multivariable limits that don't exist, path testing is often faster. For rational functions where you can factor and cancel, direct substitution works. Know when to use each method.

These five problems cover the standard scenarios you'll see in multivariable calculus courses. Work through each one until the process feels automatic. The pattern is always the same: find bounds, check convergence, conclude the limit.