Calculus worksheet - limits and functions - Free Printable

Let’s solve each limit problem step by step.

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1) Evaluate
\[
\lim_{x \to 2} \frac{x^2 + 5x + 6}{x + 2}
\]

Step 1: Try direct substitution.

Plug in \( x = 2 \):

Numerator: \( 2^2 + 5(2) + 6 = 4 + 10 + 6 = 20 \)
Denominator: \( 2 + 2 = 4 \)

So, \( \frac{20}{4} = 5 \). No indeterminate form.

✔ Answer: \( \boxed{5} \)

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2) Evaluate
\[
\lim_{x \to 2} \frac{x^3 - 8}{x - 2}
\]

Step 1: Direct substitution gives \( \frac{0}{0} \) — indeterminate form.

We factor the numerator using the difference of cubes:

\[
x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)
\]

So,

\[
\frac{x^3 - 8}{x - 2} = \frac{(x - 2)(x^2 + 2x + 4)}{x - 2}
\]

For \( x \ne 2 \), we can cancel \( x - 2 \):

\[
= x^2 + 2x + 4
\]

Now take the limit:

\[
\lim_{x \to 2} (x^2 + 2x + 4) = 4 + 4 + 4 = 12
\]

✔ Answer: \( \boxed{12} \)

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3) Evaluate
\[
\lim_{x \to \infty} \frac{x^2 + 1}{1 + 2x^3}
\]

Step 1: Compare degrees.

Numerator degree: 2
Denominator degree: 3

Since denominator grows faster than numerator as \( x \to \infty \), the limit is 0.

Formal method: Divide numerator and denominator by highest power of \( x \) in denominator → \( x^3 \):

\[
\frac{\frac{x^2}{x^3} + \frac{1}{x^3}}{\frac{1}{x^3} + 2} = \frac{\frac{1}{x} + \frac{1}{x^3}}{\frac{1}{x^3} + 2}
\]

As \( x \to \infty \), all terms with \( x \) in denominator go to 0:

\[
\frac{0 + 0}{0 + 2} = \frac{0}{2} = 0
\]

✔ Answer: \( \boxed{0} \)

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4) Evaluate
\[
\lim_{x \to \infty} \frac{4x^2 - 3x}{5x^2 - 2}
\]

Step 1: Degrees are equal (both 2).

Limit is ratio of leading coefficients.

Leading coefficient of numerator: 4
Leading coefficient of denominator: 5

So, limit = \( \frac{4}{5} \)

Formal method: Divide numerator and denominator by \( x^2 \):

\[
\frac{4 - \frac{3}{x}}{5 - \frac{2}{x^2}}
\]

As \( x \to \infty \), \( \frac{3}{x} \to 0 \), \( \frac{2}{x^2} \to 0 \):

\[
\frac{4 - 0}{5 - 0} = \frac{4}{5}
\]

✔ Answer: \( \boxed{\frac{4}{5}} \)

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## ✔ Final Answers:

1) \( \boxed{5} \)
2) \( \boxed{12} \)
3) \( \boxed{0} \)
4) \( \boxed{\frac{4}{5}} \)