5 Basic Problems on Limits just to refresh your mind.
Problem 1
Find the limit:\( \lim_{x \to 2} (3x - 4) \)
Solution:To solve this limit, we substitute the value of \(x\) directly because the function is continuous at \(x = 2\).
\[ \lim_{x \to 2} (3x - 4) = 3(2) - 4 = 6 - 4 = 2 \]
Problem 2
Find the limit:\( \lim_{x \to 0} \frac{\sin x}{x} \)
Solution:This is a standard limit. The result is known and is based on the squeeze theorem:
\[ \lim_{x \to 0} \frac{\sin x}{x} = 1 \]
Problem 3
Find the limit:\( \lim_{x \to 3} \frac{x^2 - 9}{x - 3} \)
Solution:This function is not defined at \(x = 3\), but we can simplify the expression:
\[ \lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} = \lim_{x \to 3} (x + 3) = 6 \]
Problem 4
Find the limit:\( \lim_{x \to \infty} \frac{2x^2 + 3}{x^2 - 5} \)
Solution:For limits at infinity, divide every term by the highest power of \(x\) in the denominator:
\[ \lim_{x \to \infty} \frac{2x^2 + 3}{x^2 - 5} = \lim_{x \to \infty} \frac{2 + \frac{3}{x^2}}{1 - \frac{5}{x^2}} = \frac{2}{1} = 2 \]
Problem 5
Find the limit:\( \lim_{x \to -\infty} \frac{4x^3 - x}{2x^3 + 5x^2} \)
Solution:Again, divide every term by the highest power of \(x\) in the denominator:
\[ \lim_{x \to -\infty} \frac{4x^3 - x}{2x^3 + 5x^2} = \lim_{x \to -\infty} \frac{4 - \frac{1}{x^2}}{2 + \frac{5}{x}} = \frac{4}{2} = 2 \]
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