Limits at infinity

No matter why kinds of Limits you’re looking for,
to understand it better,
the best way is to read the Step-by-Step Solution from Symbolab:
Limit Calculator from Symbolab.)

Rational functions

The KEY point is to look at the powers & coefficients of Numerator & Dominator.
Just the same with Finding the Asymptote.

Refer to previous note on the How to find Asymptote.

Example

Solve:

Quotients with square roots

The KEY point is to calculate both numerator & dominator, then calculate the limit of EACH term with in the square root.

Example

Solve:
Refer to Symbolab step-by-step solution.)

• Divide by highest dominator power to get:

• Calculate separately the limit of Numerator & Dominator:

• Calculate the Square root: Need to find limits for EACH term inside the square root.

• Then get the result easily.

Quotients with trig

The KEY point is to apply the Squeeze theorem, and it is a MUST.

Example

Solve:

• Know that -1 ≦ cos(x) ≦ 1, so we can tweak it to apply the squeeze theorem to get its limit.
• Make the inequality to: 3/-1 ≦ 3/cos(x)/-1 ≦ 3/1
• Get that right side 3/-1 = -1 and left side 3/1 =1 is not equal.
• So the limit doesn’t exist.

Easier solution steps:

• Know the inequality -1 ≦ cos(x) ≦ 1
• Replace cos(x) to ±1 in the equation, 3/±1.
• Calculate limits of two sides.
• If the results are exactly the same, then the limit is the result; Otherwise the limit doesn’t exist.

Example

Solve:

• Know that -1 ≦ sin(x) ≦ 1
• Replace sin(x) as ±1
• Left side becomes (5x+1)/(x-5), right side becomes (5x-1)/(x-5)
• Both sides’ limits are 5, so the limit exists, and is 5.